Relationship Among Three Inradii in a Right Triangle

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2017-08-15 22:50

This example illustrates the relationship among the inradii of three different triangles within a right triangle. Key mathematical concepts include geometric mean and circles. Using GeoGebra commands such as semicircle, point, segment, angle, line, orthogonal line, intersect, rotate, ray, polygon, dilate, minimum, circle arc, vector, translate, midpoint, circle sector, and more, the dynamic relationship is demonstrated. Suitable for middle school and above students to learn geometric construction and properties of circles.

GeoGebra Commands & Parameters

Semicircle(A, B) → c
Point(c) → C
Segment(A, C) → f
Segment(C, B) → g
Segment(B, A) → h
Angle(A, C, B) → α
Line(C, A) → i
Line(C, B) → j
Line(A, B) → k
OrthogonalLine(C, k) → l
Intersect(k, l) → D
OrthogonalLine(D, j) → m
OrthogonalLine(D, i) → n
Intersect(j, m) → E
Intersect(i, n) → F
Angle(D, E, B) → β
Angle(A, F, D) → γ
Line(F, E) → p
Angle(B, A, C) → δ
Rotate(B, δ / 2, A) → B'
Angle(B, A, B') → ε
Angle(C, B, A) → ζ
Rotate(C, ζ / 2, B) → C'
Angle(C, B, C') → η
Line(D, E) → q
Line(D, F) → r
Ray(B, C') → s
Intersect(q, s) → G
Ray(A, B') → t
Intersect(r, t) → H
Ray(B, G) → a
Line(D, a) → b
Intersect(i, b) → I
Ray(A, H) → d
Intersect(b, d) → J
Line(D, t) → e
Intersect(e, s) → K
Angle(F, E, D) → θ
Angle(D, F, E) → λ
Rotate(E, (-(λ / 2)), F) → E'
Angle(E', F, E) → ω
Ray(F, E') → f_1
Rotate(F, θ / 2, E) → F'
Angle(F, E, F') → κ
Ray(E, F') → g_1
Intersect(f_1, g_1) → L
Angle(C, D, A) → μ
Polygon(A, C, B) → poly1, b_1, a_1, c_1
Dilate(C, If[0 < o ≤ 1, 1 - o, 0], D) → C'_1
Segment(C, C'_1) → h_1
Dilate(D, If[1 < o ≤ 2, 2 - o, 0], F) → D'
Dilate(D, If[1 < o ≤ 2, 2 - o, 0], E) → D'_1
Segment(D, D'_1) → i_1
Segment(D, D') → j_1
Segment(A, J) → k_1
Segment(D, J) → l_1
Segment(D, K) → m_1
Segment(B, K) → n_1
Segment(F, L) → p_1
Segment(E, L) → q_1
Min({k_1, l_1, m_1, n_1, q_1, p_1}) → u
Circle(A, (0.4 * u)) → d_1
Circle(D, (0.4 * u)) → e_1
Circle(B, (0.4 * u)) → r_1
Circle(E, (0.4 * u)) → s_1
Circle(F, (0.4 * u)) → t_1
Intersect(d_1, k) → M, N
Intersect(d_1, k_1) → O,
Intersect(e_1, l_1) → P,
Intersect(e_1, k) → Q, R
Intersect(e_1, m_1) → S,
Intersect(r_1, k) → T, U
Intersect(r_1, n_1) → V,
Intersect(t_1, j_1, 1) → W
Intersect(t_1, p_1, 1) → Z
Intersect(s_1, p, 1) → A_1
Intersect(s_1, q_1, 1) → B_1
CircleSector(A, N, O) → c_2
Intersect(e_1, j_1, 1) → C_1
CircleSector(D, C_1, P) → d_2
Dilate(A, If[2 < o ≤ 3, 3 - o, 0], J) → A'
Dilate(D, If[2 < o ≤ 3, 3 - o, 0], J) → D'_2
Segment(A, A') → f_2
Segment(D, D'_2) → g_2
Dilate(D, If[3 < o ≤ 4, 4 - o, 0], K) → D'_3
Dilate(B, If[3 < o ≤ 4, 4 - o, 0], K) → B'_1
Segment(D, D'_3) → h_2
Segment(B, B'_1) → i_2
Angle(F, D, P) → ϕ
Rotate(d_2, If[4 < o ≤ 5, ((o - 4) * ϕ), ϕ], D) → e_2
Rotate(c_2, If[4 < o ≤ 5, ((o - 4) * ε), ε], A) → k_2
Dilate(D, If[5 < o ≤ 6, 6 - o, 0], B) → D'_4
Vector(D, D'_4) → v
Dilate(A, If[5 < o ≤ 6, 6 - o, 0], D) → A'_1
Vector(A, A'_1) → w
Translate(k_2, w) → p_2
Translate(e_2, v) → q_2
Dilate(F, If[6 < o ≤ 7, 7 - o, 0], E) → F'_1
Segment(F, F'_1) → j_2
CircleSector(F, W, Z) → r_2
CircleSector(E, A_1, B_1) → s_2
Dilate(F, If[7 < o ≤ 8, 8 - o, 0], L) → F'_2
Segment(F, F'_2) → l_2
Dilate(E, If[8 < o ≤ 9, 9 - o, 0], L) → E'_1
Segment(E, E'_1) → m_2
Rotate(r_2, If[9 < o ≤ 10, ((o - 9) * ω), ω], F) → t_2
Rotate(s_2, If[10 < o ≤ 11, ((o - 10) * κ), κ], E) → c_3
OrthogonalLine(K, j) → n_2
OrthogonalLine(J, i) → a_2
OrthogonalLine(L, j_2) → b_2
Intersect(j, n_2) → D_1
Intersect(b_2, j_2) → E_1
Intersect(a_2, i) → F_1
Angle(A, F_1, J) → ι
Angle(F, E_1, L) → ν
Angle(E, D_1, K) → ξ
Dilate(J, If[11 < o ≤ 12, 12 - o, 0], F_1) → J'
Dilate(K, If[11 < o ≤ 12, 12 - o, 0], D_1) → K'
Dilate(L, If[11 < o ≤ 12, 12 - o, 0], E_1) → L'
Segment(J, J') → f_3
Segment(L, L') → g_3
Segment(K, K') → h_3
Circle(L, L') → d_3
Circle(K, K') → e_3
Circle(J, J') → k_3
Dilate(F, If[12 < o ≤ 13, 13 - o, 0], C) → F'_3
Vector(F_1, F'_3) → u_1
Translate(J, u_1) → J'_1
Translate(J', u_1) → J''
Segment(J'_1, J'') → i_3
Dilate(D_1, If[13 < o ≤ 14, 14 - o, 0], J'_1) → G_1
Vector(D_1, G_1) → v_1
Translate(K, v_1) → K'_1
Translate(K', v_1) → K''
Segment(K'_1, K'') → j_3
Rotate(K'_1, If[14 < o ≤ 15, ((o - 14) * π / 2), π / 2], J'_1) → H_1
Rotate(K'', If[14 < o ≤ 15, ((o - 14) * π / 2), π / 2], J'_1) → K'''
Segment(H_1, K''') → l_3
Midpoint(C, H_1) → I_1
Rotate(H_1, If[15 < o ≤ 16, ((o - 15) * π), π], I_1) → J_1
Semicircle(C, H_1) → p_3
OrthogonalLine(J'_1, j) → m_3
Intersect(p_3, m_3, 1) → K_1
CircleSector(I_1, H_1, J_1) → q_3
CircleArc(I_1, H_1, J_1) → r_3
Angle(K_1, J'_1, C) → ο
Dilate(K_1, If[17 < o ≤ 18, 18 - o, 0], J'_1) → L_1
Segment(K_1, L_1) → n_3
Line(J'_1, g_3) → s_3
Intersect(q_3, r_3, 1) → M_1
Intersect(r_3, s_3) → N_1, R_1
Dilate(E_1, If[18 < o ≤ 19, 19 - o, 0], J'_1) → O_1
Vector(E_1, O_1) → w_1
Translate(L, w_1) → L'_1
Translate(L', w_1) → L''
Segment(L'_1, L'') → t_3
Point(s_3) → P_1
Angle(P_1, J'_1, K_1) → σ
Rotate(L'_1, If[19 < o ≤ 20, ((o - 19) * σ), σ], J'_1) → Q_1
Rotate(L'', If[19 < o ≤ 20, ((o - 19) * σ), σ], J'_1) → L'''
Segment(Q_1, L''') → a_3

AI Drawing Prompt Reference

Draw a right triangle and inside it, illustrate three smaller triangles with their incircles to show their relationships. Start by constructing the right triangle and draw a semicircle with the hypotenuse as the diameter. From the intersection points of the semicircle with the legs, draw perpendicular lines to the hypotenuse, and then from these points, draw perpendicular lines to the legs to find the vertices of the smaller triangles. Construct the incircles of these smaller triangles and label the key angles to demonstrate the geometric mean relationships among the inradii.

The prompt above was compiled from the original drawing commands and is for reference only.