💡Okay, so I have this problem here about polygons. It says that the sum of the interior angles of a polygon is three times the sum of its exterior angles, and I need to find the number of sides. The options are 6, 7, 8, or 9. Hmm, let's see how I can approach this.First, I remember that for any polygon, the sum of the exterior angles is always 360 degrees. That's a key point. I think it doesn't matter how many sides the polygon has; the sum of all exterior angles is always 360 degrees. So, that's one piece of information I can use.Now, the sum of the interior angles is given by a formula. I think it's something like (n-2) multiplied by 180 degrees, where n is the number of sides. So, if I denote the number of sides as n, then the sum of the interior angles would be 180(n-2) degrees.The problem states that the sum of the interior angles is three times the sum of the exterior angles. Since the sum of the exterior angles is 360 degrees, three times that would be 3 * 360, which is 1080 degrees. So, the sum of the interior angles is 1080 degrees.Putting this together, I can set up an equation:180(n - 2) = 1080Now, I need to solve for n. Let's do that step by step.First, expand the left side:180n - 360 = 1080Next, I'll add 360 to both sides to isolate the term with n:180n = 1080 + 360180n = 1440Now, divide both sides by 180 to solve for n:n = 1440 / 180n = 8So, the number of sides is 8. Let me double-check to make sure I didn't make any mistakes.If n = 8, then the sum of the interior angles should be 180(8-2) = 180 * 6 = 1080 degrees. The sum of the exterior angles is 360 degrees, and three times that is indeed 1080 degrees. So, it checks out.Therefore, the correct answer should be 8, which is option C.

💡First, I need to determine the number of students below 8 years of age. Since 20% of the students are below 8 years old and the total number of students is 150, I can calculate this by multiplying 150 by 20%.Next, I know the number of students who are exactly 8 years old is 72. To find out how many students are above 8 years old, I need to subtract the number of students below 8 and those who are exactly 8 from the total number of students.Finally, I'll perform the calculations to find the total number of students above 8 years of age.

💡First, I need to determine how many seeds Nancy planted in the big garden. She planted all 17 cucumber seeds and 24 tomato seeds there.Next, I'll subtract the seeds planted in the big garden from the total seeds to find out how many seeds are left for the small gardens.For the small gardens, each requires 2 tomato seeds, 1 capsicum seed, and 1 cucumber seed. However, since all cucumber seeds have been used in the big garden, I'll focus on the remaining tomato and capsicum seeds.The number of small gardens will be limited by the type of seed with the smallest remaining quantity after planting in the big garden. In this case, it's the tomato seeds with 18 left. Since each small garden requires 2 tomato seeds, I'll divide the remaining tomato seeds by 2 to find the number of small gardens.

💡Alright, so I have this problem here where I need to find the length of the segment AB, where A and B are the intersection points between a line l and a curve C. The equations are given in parametric form, which I remember is a way to express the coordinates of a point using a parameter. Let me try to break this down step by step.First, let me write down the parametric equations for both the line l and the curve C so I can see them clearly.For line l:- x = 1 + s- y = 1 - sHere, s is the parameter. So, as s changes, the point (x, y) moves along the line.For curve C:- x = t + 2- y = t²Here, t is the parameter. So, as t changes, the point (x, y) moves along the curve, which seems to be a parabola since y is t squared.My goal is to find where these two meet, meaning I need to find the values of s and t where both x and y are the same for both equations. Once I have those points, I can calculate the distance between them.So, let me set the x's equal and the y's equal because at the intersection points, both x and y must be the same for both equations.From line l, x = 1 + s and y = 1 - s.From curve C, x = t + 2 and y = t².So, setting x's equal:1 + s = t + 2And setting y's equal:1 - s = t²Now, I have two equations:1. 1 + s = t + 22. 1 - s = t²I need to solve these two equations for s and t. Let me see how I can do this.From equation 1, I can solve for s in terms of t:1 + s = t + 2Subtract 1 from both sides:s = t + 1So, s is equal to t + 1. Now, I can substitute this expression for s into equation 2.Equation 2 is:1 - s = t²Substituting s = t + 1:1 - (t + 1) = t²Simplify the left side:1 - t - 1 = t²Which simplifies to:-t = t²Let me rewrite that:t² + t = 0Factor out a t:t(t + 1) = 0So, this gives two solutions:t = 0 or t + 1 = 0Which means:t = 0 or t = -1Alright, so t can be 0 or -1. Now, let's find the corresponding s values using s = t + 1.For t = 0:s = 0 + 1 = 1For t = -1:s = -1 + 1 = 0So, we have two sets of parameters:1. t = 0, s = 12. t = -1, s = 0Now, let's find the coordinates of points A and B using these parameters.First, for t = 0:From curve C:x = 0 + 2 = 2y = 0² = 0So, point A is (2, 0)From line l, with s = 1:x = 1 + 1 = 2y = 1 - 1 = 0Consistent, so point A is indeed (2, 0)Next, for t = -1:From curve C:x = -1 + 2 = 1y = (-1)² = 1So, point B is (1, 1)From line l, with s = 0:x = 1 + 0 = 1y = 1 - 0 = 1Consistent, so point B is indeed (1, 1)So, the two intersection points are A(2, 0) and B(1, 1). Now, I need to find the distance between these two points.The distance formula between two points (x1, y1) and (x2, y2) is:Distance = sqrt[(x2 - x1)² + (y2 - y1)²]Plugging in the coordinates of A and B:Distance AB = sqrt[(1 - 2)² + (1 - 0)²]= sqrt[(-1)² + (1)²]= sqrt[1 + 1]= sqrt[2]So, the length of |AB| is sqrt(2).Wait, let me double-check my calculations to make sure I didn't make any mistakes.First, solving for t:From x's: 1 + s = t + 2 => s = t + 1From y's: 1 - s = t²Substituting s: 1 - (t + 1) = t² => -t = t² => t² + t = 0 => t(t + 1) = 0 => t = 0 or t = -1That seems correct.Then, s values:t = 0 => s = 1t = -1 => s = 0Coordinates:For t = 0: x = 2, y = 0For t = -1: x = 1, y = 1From line l:s = 1: x = 2, y = 0s = 0: x = 1, y = 1Consistent.Distance calculation:Difference in x: 1 - 2 = -1Difference in y: 1 - 0 = 1Squares: (-1)^2 = 1, (1)^2 = 1Sum: 1 + 1 = 2Square root: sqrt(2)Everything checks out. So, the length of |AB| is indeed sqrt(2).I think that's it. I don't see any errors in my reasoning or calculations.**Final Answer**The length of |AB| is boxed{sqrt{2}}.

💡Alright, so I have this geometry problem to prove that in a triangle, an angle bisector divides the angle between the altitude and the radius of the circumscribed circle, drawn from the same vertex, into two equal parts. Hmm, that sounds a bit complicated, but let's break it down step by step.First, let me visualize the triangle. Let's say we have triangle ABC, with vertex A. From A, we can draw an altitude to the opposite side BC, which I'll call AD, where D is the foot of the altitude on BC. Then, we also have the radius of the circumscribed circle (circumradius) from A to the circumcenter O. So, AO is the radius of the circumcircle.Now, the problem is asking about the angle between AD (the altitude) and AO (the circumradius). Specifically, it wants to show that the angle bisector of angle A divides this angle into two equal parts. So, if we draw the angle bisector of angle A, let's call it AE, where E is some point on BC, then AE should split the angle between AD and AO into two equal angles.Okay, so to rephrase, I need to show that angle DAE is equal to angle EAO. That is, AE bisects the angle between AD and AO.Let me recall some properties of triangle centers and angle bisectors. The circumradius AO is related to the sides of the triangle and the angles. The altitude AD is related to the area of the triangle. The angle bisector AE divides the opposite side BC in the ratio of the adjacent sides.Maybe I can use trigonometric relationships or properties of similar triangles here. Let me consider the triangle AOD and triangle AOE. If I can show that these triangles are similar or have some proportional sides, that might help.Wait, but O is the circumcenter, so it's the intersection point of the perpendicular bisectors of the triangle. That means AO is not only a radius but also a perpendicular bisector of BC if the triangle is isosceles, but in general, it's just a perpendicular bisector in the case of an isosceles triangle. Hmm, no, actually, in any triangle, the circumcenter is equidistant from all three vertices, so AO = BO = CO.Maybe I can use the fact that AO is equal to BO and CO. Let me think about the angles involved. Since AE is the angle bisector, it divides angle BAC into two equal angles. So, angle BAE equals angle EAC.Now, considering the altitude AD, it creates two right angles at D. So, angle ADB and angle ADC are both 90 degrees. Maybe I can relate the angles in triangles ABD and ACD with the angles involving AO and AE.Alternatively, perhaps I can use coordinates to model this problem. Let me place triangle ABC on a coordinate plane to make things more concrete. Let's set point A at (0, 0), point B at (c, 0), and point C at (d, e). Then, I can find the coordinates of D (the foot of the altitude from A), O (the circumcenter), and E (the point where the angle bisector meets BC).But before diving into coordinates, maybe there's a synthetic geometry approach. Let me recall that the circumradius AO can be expressed in terms of the sides of the triangle and the angles. Specifically, the formula for the circumradius is R = a / (2 sin A), where a is the length of the side opposite angle A.Also, the length of the altitude AD can be expressed as AD = b sin C = c sin B, where b and c are the lengths of the sides adjacent to angle A.Hmm, perhaps I can relate these lengths to the angles involved. Since AE is the angle bisector, it divides angle A into two equal parts, so maybe the angles between AE and AD, and between AE and AO, can be related through some trigonometric identities.Wait, another thought: maybe I can use the fact that in triangle AOE and triangle AOD, certain angles are equal or sides are proportional. But I need to be careful about which triangles I'm comparing.Alternatively, perhaps considering the properties of the Euler line, but I don't think that's directly applicable here since we're dealing with the circumradius and an angle bisector.Let me try to draw this out mentally. From vertex A, we have three lines: the altitude AD, the circumradius AO, and the angle bisector AE. The problem states that AE bisects the angle between AD and AO. So, if I can show that the angles between AD and AE, and between AE and AO, are equal, then I'm done.Maybe I can express these angles in terms of the angles of the triangle and show that they are equal. Let's denote angle BAD as α and angle BAC as 2α since AE is the angle bisector. Wait, no, angle BAC is being bisected by AE, so if angle BAC is θ, then angle BAE and angle EAC are both θ/2.But how does this relate to the angles involving AD and AO? Let me think about the angles at point A.Since AD is the altitude, angle ADB is 90 degrees. So, angle BAD is complementary to angle ABD. Similarly, angle CAD is complementary to angle ACD.But AO is the circumradius, so it relates to the circumcircle. Maybe I can use the fact that angle AOB is twice angle ACB, as the central angle is twice the inscribed angle.Wait, that might be useful. Let me recall that in the circumcircle, the central angle subtended by an arc is twice the inscribed angle subtended by the same arc. So, angle AOB = 2 angle ACB.Similarly, angle AOC = 2 angle ABC.Hmm, so if I can relate angle AOB to the angles in the triangle, maybe I can find a relationship between AO and the other elements.But I'm not sure yet how to connect this to the angle bisector AE. Maybe I need to consider triangle AOE and triangle AOD.Alternatively, perhaps using vectors or coordinate geometry would make this clearer. Let me try setting up coordinates.Let me place point A at the origin (0,0). Let me assume that side BC is horizontal for simplicity, so point D, the foot of the altitude from A, is at (d, 0). Then, point B can be at (b, 0) and point C at (c, 0), but wait, that would make BC a straight line, which is not possible unless it's a degenerate triangle. Hmm, maybe I need to adjust.Wait, no, if I place A at (0,0), and since AD is the altitude, D is at (d, 0), and then B and C are on the line perpendicular to AD at D. Wait, no, AD is the altitude, so BC is the base, and AD is perpendicular to BC.So, let me set point A at (0, h), and BC on the x-axis from (b, 0) to (c, 0). Then, the foot of the altitude D is at (d, 0). So, AD is the line from (0, h) to (d, 0).Now, the circumradius AO goes from A(0, h) to the circumcenter O. The circumcenter is the intersection of the perpendicular bisectors of the sides. So, to find O, I need to find the perpendicular bisectors of AB and AC.The midpoint of AB is ((b/2), h/2), and the slope of AB is (0 - h)/(b - 0) = -h/b. So, the perpendicular bisector of AB has slope b/h and passes through (b/2, h/2).Similarly, the midpoint of AC is ((c/2), h/2), and the slope of AC is (0 - h)/(c - 0) = -h/c. So, the perpendicular bisector of AC has slope c/h and passes through (c/2, h/2).Now, the circumcenter O is the intersection of these two perpendicular bisectors. Let me find the equations of these bisectors.For the perpendicular bisector of AB:Slope = b/hPoint: (b/2, h/2)Equation: y - h/2 = (b/h)(x - b/2)Similarly, for the perpendicular bisector of AC:Slope = c/hPoint: (c/2, h/2)Equation: y - h/2 = (c/h)(x - c/2)Now, to find O, solve these two equations simultaneously.Let me write them out:1. y = (b/h)(x - b/2) + h/22. y = (c/h)(x - c/2) + h/2Set them equal:(b/h)(x - b/2) + h/2 = (c/h)(x - c/2) + h/2Subtract h/2 from both sides:(b/h)(x - b/2) = (c/h)(x - c/2)Multiply both sides by h:b(x - b/2) = c(x - c/2)Expand:bx - b²/2 = cx - c²/2Bring all terms to one side:bx - cx = b²/2 - c²/2Factor:x(b - c) = (b² - c²)/2Note that b² - c² = (b - c)(b + c), so:x(b - c) = (b - c)(b + c)/2Assuming b ≠ c (otherwise, the triangle is isosceles and the circumcenter lies on the altitude), we can divide both sides by (b - c):x = (b + c)/2Now, plug this back into one of the equations to find y. Let's use equation 1:y = (b/h)( (b + c)/2 - b/2 ) + h/2Simplify inside the parentheses:(b + c)/2 - b/2 = c/2So,y = (b/h)(c/2) + h/2 = (bc)/(2h) + h/2Therefore, the circumcenter O is at ((b + c)/2, (bc)/(2h) + h/2)Now, the angle bisector AE from A(0, h) to E on BC. The coordinates of E can be found using the angle bisector theorem, which states that BE/EC = AB/AC.First, find AB and AC.AB = distance from A(0, h) to B(b, 0) = sqrt(b² + h²)AC = distance from A(0, h) to C(c, 0) = sqrt(c² + h²)So, BE/EC = AB/AC = sqrt(b² + h²)/sqrt(c² + h²)Let me denote BE = k and EC = m, so k/m = sqrt(b² + h²)/sqrt(c² + h²)Also, since BE + EC = BC = |c - b|, we have k + m = |c - b|Let me solve for k and m.Let me set k = (sqrt(b² + h²)/(sqrt(b² + h²) + sqrt(c² + h²))) * |c - b|Similarly, m = (sqrt(c² + h²)/(sqrt(b² + h²) + sqrt(c² + h²))) * |c - b|Assuming c > b for simplicity, so BC = c - b.Thus, E is located at (b + k, 0) = (b + (sqrt(b² + h²)/(sqrt(b² + h²) + sqrt(c² + h²)))*(c - b), 0)Now, the coordinates of E are:E_x = b + (sqrt(b² + h²)/(sqrt(b² + h²) + sqrt(c² + h²)))*(c - b)E_y = 0Now, we have points A(0, h), O((b + c)/2, (bc)/(2h) + h/2), and E(E_x, 0)We need to find the angles between AD, AE, and AO.First, let's find the vectors for these lines.Vector AD goes from A(0, h) to D(d, 0). So, vector AD is (d, -h)Vector AE goes from A(0, h) to E(E_x, 0). So, vector AE is (E_x, -h)Vector AO goes from A(0, h) to O((b + c)/2, (bc)/(2h) + h/2). So, vector AO is ((b + c)/2, (bc)/(2h) + h/2 - h) = ((b + c)/2, (bc)/(2h) - h/2)Now, to find the angles between these vectors, we can use the dot product formula:cos(theta) = (u . v)/(|u||v|)So, let's compute the angles between AD and AE, and between AE and AO.First, angle between AD and AE:Vector AD: (d, -h)Vector AE: (E_x, -h)Dot product: d*E_x + (-h)*(-h) = d*E_x + h²Magnitude of AD: sqrt(d² + h²)Magnitude of AE: sqrt(E_x² + h²)So, cos(theta1) = (d*E_x + h²)/(sqrt(d² + h²)*sqrt(E_x² + h²))Similarly, angle between AE and AO:Vector AE: (E_x, -h)Vector AO: ((b + c)/2, (bc)/(2h) - h/2)Dot product: E_x*(b + c)/2 + (-h)*((bc)/(2h) - h/2) = (E_x*(b + c))/2 + (-h)*(bc/(2h) - h/2)Simplify the second term:(-h)*(bc/(2h) - h/2) = (-h)*(bc/(2h)) + (-h)*(-h/2) = -bc/2 + h²/2So, total dot product: (E_x*(b + c))/2 - bc/2 + h²/2Magnitude of AE: sqrt(E_x² + h²)Magnitude of AO: sqrt( ((b + c)/2)^2 + ((bc)/(2h) - h/2)^2 )So, cos(theta2) = [ (E_x*(b + c))/2 - bc/2 + h²/2 ] / [ sqrt(E_x² + h²) * sqrt( ((b + c)/2)^2 + ((bc)/(2h) - h/2)^2 ) ]Now, to show that theta1 = theta2, we need to show that cos(theta1) = cos(theta2). That would imply that the angles are equal.This seems quite involved. Maybe there's a simplification or a relationship between the variables that I can exploit.Wait, I know that D is the foot of the altitude from A to BC, so AD is perpendicular to BC. Therefore, the slope of AD is (0 - h)/(d - 0) = -h/d, and the slope of BC is (0 - 0)/(c - b) = 0, since BC is on the x-axis. Wait, no, BC is from (b, 0) to (c, 0), so it's horizontal, hence slope 0. Therefore, the altitude AD is vertical if BC is horizontal? Wait, no, AD is perpendicular to BC, which is horizontal, so AD is vertical. Therefore, d should be equal to the x-coordinate of A projected onto BC. Wait, but A is at (0, h), so the foot D would be at (0, 0) if BC is on the x-axis. Wait, that can't be right because then D would coincide with the origin, but BC is from (b, 0) to (c, 0), so unless b or c is 0, D is not at (0,0).Wait, I think I made a mistake in placing the coordinates. If I place A at (0, h), and BC on the x-axis from (b, 0) to (c, 0), then the foot of the altitude from A to BC is at (d, 0), where d is the x-coordinate where the vertical line from A meets BC. But since BC is horizontal, the altitude from A is vertical, so d must be equal to the x-coordinate of A, which is 0. Therefore, D is at (0, 0). But then BC is from (b, 0) to (c, 0), so unless b or c is 0, D is not on BC. Wait, that can't be. So, perhaps my coordinate system is flawed.Let me rethink the coordinate system. Maybe I should place BC on the x-axis with B at (-k, 0) and C at (k, 0) to make it symmetric, and A at (0, h). Then, the foot of the altitude D would be at (0, 0), which is the midpoint of BC if the triangle is isosceles, but in general, it's not necessarily the midpoint.Wait, no, if BC is from (-k, 0) to (k, 0), then the midpoint is (0,0), and if the triangle is isosceles with AB=AC, then the altitude from A would indeed be at (0,0). But in a general triangle, the foot of the altitude D would not necessarily be at (0,0). Hmm, maybe this symmetric approach complicates things.Alternatively, perhaps I should place point A at (0,0), point B at (c,0), and point C at (d,e). Then, the altitude from A would be along the y-axis if BC is horizontal, but that might not necessarily be the case.Wait, maybe I'm overcomplicating this with coordinates. Let me try a synthetic approach again.Since AO is the circumradius, and AD is the altitude, and AE is the angle bisector, perhaps I can use properties of these lines in relation to the triangle's angles.I know that the angle bisector divides the opposite side in the ratio of the adjacent sides. So, AE divides BC into segments proportional to AB and AC.Also, the circumradius AO relates to the sides and angles via the formula a = 2R sin A, where a is the side opposite angle A.The altitude AD can be expressed as AD = b sin C = c sin B.Maybe I can relate these expressions to the angles between AD, AE, and AO.Alternatively, perhaps considering the triangle formed by points A, O, and D. In triangle AOD, AO is the circumradius, AD is the altitude, and OD is some segment. Maybe I can find some relationship in this triangle.Wait, but I don't know much about OD. Maybe I can express OD in terms of other known quantities.Alternatively, perhaps using trigonometric identities in triangle AOE and triangle AOD.Wait, another idea: since AE is the angle bisector, it divides angle A into two equal parts. So, angle BAE = angle EAC = θ/2, where θ is angle BAC.Now, if I can express the angles between AD and AE, and between AE and AO, in terms of θ and other known angles, maybe I can show they are equal.Let me denote angle between AD and AE as φ, and angle between AE and AO as ψ. We need to show that φ = ψ.So, φ = angle between AD and AEψ = angle between AE and AOIf I can express φ and ψ in terms of θ and other angles, perhaps I can show they are equal.Alternatively, perhaps considering the fact that AO is the circumradius, and AD is the altitude, and AE is the angle bisector, there might be some known theorem or property that relates these.Wait, I recall that in a triangle, the angle between the altitude and the angle bisector can be related to other elements, but I'm not sure about the circumradius.Alternatively, maybe using the fact that the circumradius AO makes an angle with the sides AB and AC that is related to the angles of the triangle.Wait, since AO is the circumradius, the angles between AO and AB, and AO and AC, are related to the angles of the triangle.Specifically, in triangle AOB, which is isosceles with AO = BO, the base angles are equal. So, angle OAB = angle OBA.Similarly, in triangle AOC, which is isosceles with AO = CO, the base angles are equal. So, angle OAC = angle OCA.Wait, that might be useful. Let me denote angle OAB as α and angle OAC as β.Since triangle AOB is isosceles with AO = BO, angle OAB = angle OBA = α.Similarly, triangle AOC is isosceles with AO = CO, so angle OAC = angle OCA = β.Now, in triangle ABC, the sum of angles at B and C is 180 - angle A.But angle OBA = α and angle OCA = β, so angle ABC = angle OBA + angle OBC, but I'm not sure.Wait, perhaps I can relate α and β to the angles of the triangle.Wait, in triangle ABC, angle ABC is equal to angle OBA + angle OBC, but I don't know angle OBC.Alternatively, since O is the circumcenter, the angles at O are related to the central angles.Wait, angle AOB is equal to 2 angle ACB, as the central angle is twice the inscribed angle.Similarly, angle AOC is equal to 2 angle ABC.So, angle AOB = 2 angle ACBangle AOC = 2 angle ABCNow, in triangle AOB, which is isosceles with AO = BO, the base angles are equal, so angle OAB = angle OBA = α.Similarly, in triangle AOC, which is isosceles with AO = CO, the base angles are equal, so angle OAC = angle OCA = β.Now, in triangle ABC, angle ABC = angle OBA + angle OBC = α + angle OBCSimilarly, angle ACB = angle OCA + angle OCB = β + angle OCBBut since angle AOB = 2 angle ACB, and angle AOB is also equal to 180 - 2α (since triangle AOB has angles α, α, and 180 - 2α).Similarly, angle AOC = 2 angle ABC = 180 - 2β.Wait, let me write that down:In triangle AOB:angle AOB = 180 - 2αBut angle AOB = 2 angle ACBSo, 180 - 2α = 2 angle ACBThus, angle ACB = (180 - 2α)/2 = 90 - αSimilarly, in triangle AOC:angle AOC = 180 - 2βBut angle AOC = 2 angle ABCSo, 180 - 2β = 2 angle ABCThus, angle ABC = (180 - 2β)/2 = 90 - βNow, in triangle ABC, the sum of angles is 180:angle BAC + angle ABC + angle ACB = 180θ + (90 - β) + (90 - α) = 180θ + 180 - α - β = 180θ = α + βSo, angle BAC = α + βBut AE is the angle bisector, so it divides angle BAC into two equal parts:angle BAE = angle EAC = θ/2 = (α + β)/2Now, let's consider the angles between AD, AE, and AO.First, angle between AD and AE: let's call this φSecond, angle between AE and AO: let's call this ψWe need to show that φ = ψTo find φ and ψ, we can express them in terms of α and β.Let me consider the angles at point A.Since AD is the altitude, angle BAD is complementary to angle ABC.Similarly, angle CAD is complementary to angle ACB.Wait, let's see:In triangle ABD, which is right-angled at D, angle BAD = 90 - angle ABCSimilarly, in triangle ACD, angle CAD = 90 - angle ACBBut angle ABC = 90 - βangle ACB = 90 - αSo, angle BAD = 90 - (90 - β) = βangle CAD = 90 - (90 - α) = αTherefore, angle BAD = βangle CAD = αSince angle BAC = α + β, this makes sense.Now, AE is the angle bisector, so it divides angle BAC into two equal parts:angle BAE = angle EAC = (α + β)/2Now, let's find the angles between AD and AE, and between AE and AO.First, angle between AD and AE: φSince AD is the altitude, and AE is the angle bisector, φ is the angle between AD and AE.From above, angle BAD = β, and angle BAE = (α + β)/2Therefore, φ = angle BAD - angle BAE = β - (α + β)/2 = (2β - α - β)/2 = (β - α)/2Similarly, angle between AE and AO: ψWe need to find the angle between AE and AO.From earlier, angle OAB = α, and angle BAE = (α + β)/2Therefore, angle between AE and AO is angle OAE = angle OAB - angle BAE = α - (α + β)/2 = (2α - α - β)/2 = (α - β)/2Wait, but we have φ = (β - α)/2 and ψ = (α - β)/2These are negatives of each other, but angles are positive, so perhaps I made a miscalculation.Wait, actually, angles are directed, so depending on the orientation, the signs might differ. But in terms of magnitude, |φ| = |ψ|But we need to show that φ = ψ, not just their magnitudes are equal.Wait, perhaps I need to consider the direction of the angles.Alternatively, maybe I made a mistake in the calculation.Let me re-examine:angle BAD = βangle BAE = (α + β)/2Therefore, angle between AD and AE is φ = angle BAD - angle BAE = β - (α + β)/2 = (2β - α - β)/2 = (β - α)/2Similarly, angle between AE and AO:From point A, angle between AE and AO is angle OAE.We know that angle OAB = αangle BAE = (α + β)/2Therefore, angle OAE = angle OAB - angle BAE = α - (α + β)/2 = (2α - α - β)/2 = (α - β)/2So, φ = (β - α)/2 and ψ = (α - β)/2But these are negatives of each other. However, angles are positive, so perhaps I need to take absolute values.But in reality, depending on the position of E relative to O, the angle could be on the other side.Wait, perhaps I need to consider the orientation. If α > β, then angle OAE would be positive, and angle between AD and AE would be negative, but since angles are positive, we take the absolute value.Therefore, |φ| = |ψ|But the problem states that the angle bisector divides the angle between AD and AO into two equal parts, meaning that φ = ψ in measure, regardless of direction.Therefore, since |φ| = |ψ|, and they are adjacent angles forming the angle between AD and AO, they must be equal.Hence, the angle bisector AE divides the angle between AD and AO into two equal parts.Therefore, the proof is complete.

💡Okay, so I have this problem: I need to find the smallest perfect square larger than 1 that has a perfect square number of positive integer factors. Hmm, let me try to break this down step by step.First, let's recall what a perfect square is. A perfect square is a number that can be expressed as the square of an integer. For example, 4 is a perfect square because it's 2 squared, 9 is 3 squared, and so on. So, we're looking for a number like that, but it has to be larger than 1. That means the smallest perfect square we're considering is 4.Now, the next part is about the number of positive integer factors. The number of factors a number has is determined by its prime factorization. If a number N can be expressed as ( N = p_1^{a_1} times p_2^{a_2} times dots times p_n^{a_n} ), where ( p_1, p_2, dots, p_n ) are prime numbers and ( a_1, a_2, dots, a_n ) are their respective exponents, then the total number of positive integer factors of N is ( (a_1 + 1) times (a_2 + 1) times dots times (a_n + 1) ).So, for a perfect square, all the exponents in its prime factorization must be even numbers. For example, 4 is ( 2^2 ), so the number of factors is ( (2 + 1) = 3 ). Similarly, 9 is ( 3^2 ), so it also has 3 factors. The problem asks for a perfect square that has a perfect square number of factors. That means the number of factors itself should be a perfect square. So, we need to find the smallest perfect square (larger than 1) such that when we calculate its number of factors, that number is also a perfect square.Let me start checking perfect squares one by one:1. **4**: Prime factorization is ( 2^2 ). Number of factors is ( 2 + 1 = 3 ). 3 is not a perfect square.2. **9**: Prime factorization is ( 3^2 ). Number of factors is ( 2 + 1 = 3 ). Again, 3 isn't a perfect square.3. **16**: Prime factorization is ( 2^4 ). Number of factors is ( 4 + 1 = 5 ). 5 isn't a perfect square.4. **25**: Prime factorization is ( 5^2 ). Number of factors is ( 2 + 1 = 3 ). Still not a perfect square.5. **36**: Prime factorization is ( 2^2 times 3^2 ). Number of factors is ( (2 + 1) times (2 + 1) = 3 times 3 = 9 ). 9 is a perfect square because it's ( 3^2 ).So, 36 is the first perfect square larger than 1 that has a perfect square number of factors. Let me just verify that there isn't a smaller one that I might have missed. I checked 4, 9, 16, 25, and 36. All the ones before 36 don't satisfy the condition, so 36 must be the answer.I think that's it. 36 is the smallest perfect square larger than 1 with a perfect square number of positive integer factors.