Rearrangement Proof of the Pythagorean Theorem

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2013-05-30 05:02

In collections Pythagorean Theorem

This example demonstrates the famous 'rearrangement' proof of the Pythagorean Theorem, showing the theorem through the movement and rearrangement of shapes. The core math concepts involved are the Pythagorean Theorem and geometric transformations. Construction highlights include using commands and objects like Polygon, Midpoint, Ray, Rotate, Circle, Segment, Angle, Centroid, Line, Point, OrthogonalLine, Intersect, and Vector in GeoGebra. Suitable for middle school students and above to learn geometric proof methods.

GeoGebra Commands & Parameters

Polygon(A, C, B) → poly1, b_1, a_1, c_2
Midpoint(b_1) → M
Ray(B, M) → n_2
Rotate(B, (-(((90 * Progress_4))°)), B) → B'
Circle(B', sqrt(b^(2) + ((0.5 * a))^(2)) - 0.1) → d_3
Rotate(L, ((90 * Progress_2))°, J) → L'
Rotate(L', ((Progress_3 * 90))°, L') → L''
Circle(L'', sqrt(a^(2) + ((0.5 * b))^(2)) - 0.1) → c_3
Rotate(K, ((90 * Progress_2))°, J) → K'
Polygon(J, K', L') → poly4, l', j, k'
Midpoint(l') → Z
Ray(L', Z) → m_2
Rotate(E, ((90 * Progress))°, D) → E'
Rotate(F, ((90 * Progress))°, D) → F'
Polygon(F', D, E') → poly2, e', f', d
Rotate(H, ((90 * Progress_1))°, G) → H'
Rotate(I, ((90 * Progress_1))°, G) → I'
Polygon(H', I', G) → poly3, g, h', i'
If(Progress ≟ 0, 0, 1) → e
If(Progress_1 ≟ 0, false, true) → h
Rotate(J, ((Progress_3 * 90))°, L') → J'
Rotate(K', ((Progress_3 * 90))°, L') → K''
Polygon(J', K'', L'') → poly4', l'', j', k''
If(Progress ≟ 1, If[Progress_1 ≟ 0, true, false], true) → Progress_{visibility}
If(Progress ≟ 1, If[Progress_2 ≟ 0, true, false], false) → Progress1_{visibility}
Polygon(C, A, B) → poly5, b_2, c_1, a_2
Rotate(C, (-(((90 * Progress_4))°)), B) → C'
Rotate(A, (-(((90 * Progress_4))°)), B) → A'
Polygon(C', A', B') → poly5', b', c', a'
If(Progress_2 ≟ 0, false, If[Progress_3 ≟ 0, true, false]) → i
If(Progress_3 ≟ 0, true, false) → k
If(Progress_1 ≟ 1, If[Progress_3 ≟ 0, true, false], false) → Progress2_{visibility}
If(Progress_3 ≟ 0, false, true) → l
Segment(J, A) → m
Segment(A, D) → n
Segment(D, G) → p
Segment(G, J) → q
If(Progress_2 ≟ 1, true, false) → square
Angle(B, A, C) → α
Angle(E', D, F') → β
Angle(H', G, I') → γ
Angle(K', J, L') → δ
If(Progress ≟ 1, true, false) → Right1
If(Progress_1 ≟ 1, true, false) → Right2
Angle(E', F', C) → ε
Angle(H', I', F') → ζ
Angle(C, H', I') → η
Angle(B, C, H') → θ
If(Progress ≟ 1, If[Progress_3 ≟ 0, true, false], false) → Blue1
If(Progress_1 ≟ 1, If[Progress_3 ≟ 0, true, false], false) → Blue2
If(Progress_2 ≟ 1, If[Progress_3 ≟ 0, true, false], false) → Blue3
Polygon(C, L', E', B) → poly6, r, s, t, d_1
If(Progress_2 ≟ 1 ∧ Progress_3 ≟ 0, true, false) → Csquare
Polygon(J, L, J', L') → poly7, j_1, l_1, j'_1, l'_1
Polygon(L, A, B, A') → poly8, l_2, a_3, b_3, a'_1
Angle(J, L', J') → ι
Angle(L', J', L) → κ
Angle(J', L, J) → λ
Angle(L, A', B) → μ
Angle(A', B, A) → ν
Angle(A, L, A') → ξ
If(Progress_3 ≟ 1, true, false) → Bluesquares
Midpoint(c_2) → N
Midpoint(a_1) → O
Midpoint(e') → P
Midpoint(f') → Q
Midpoint(d) → R
Midpoint(h') → S
Midpoint(i') → T
Midpoint(g) → U
If(Progress_1 ≟ 1, true, false) → Legs_{3rd}
Midpoint(k') → V
Midpoint(j) → W
If(Progress ≟ 1, If[Progress_3 ≟ 0, true, false], false) → Hypo_{2nd}
If(Progress_1 ≟ 1, If[Progress_3 ≟ 0, true, false], false) → Hypo_{3rd}
If(Progress_2 ≟ 1, If[Progress_3 ≟ 0, true, false], false) → Hypo_{4th}
Centroid(poly6) → A_1
If(Progress ≟ 1, true, false) → Legs_1
If(Progress_2 ≟ 1, If[Progress_3 ≟ 0, true, false], false) → Legs_{4th}
Midpoint(l'_1) → B_1
Midpoint(k'') → C_1
Midpoint(j_1) → D_1
Midpoint(l_1) → E_1
Midpoint(a'_1) → F_1
Midpoint(b_3) → G_1
Midpoint(a_3) → H_1
Midpoint(l_2) → I_1
If(Progress_3 ≟ 1, true, false) → Bluelabels
If(Progress_2 ≟ 1 ∧ Progress_3 ≟ 0, true, false) → textofcsquared
Centroid(poly7) → J_1
Centroid(poly8) → K_1
If(Progress_3 ≟ 1, true, false) → aandbsquareds
Line(L_1, yAxis) → c
Line(M_1, yAxis) → e_1
Point(c) → N_1
OrthogonalLine(N_1, c) → f_1
Point(c) → O_1
OrthogonalLine(O_1, c) → g_1
Point(c) → P_1
OrthogonalLine(P_1, c) → h_1
Intersect(e_1, f_1) → Q_1
Intersect(e_1, g_1) → R_1
Intersect(e_1, h_1) → S_1
Polygon(P_1, O_1, R_1, S_1) → poly9, p_1, o_1, r_1, s_1
Polygon(O_1, N_1, Q_1, R_1) → poly10, o_2, n_1, q_1, r_2
If(Progress ≟ 0, true, false) → Title
Segment(T_1, U_1) → i_1
Segment(U_1, V_1) → k_1
Segment(V_1, W_1) → m_1
Segment(W_1, T_1) → t_1
If(Progress ≟ 0, true, false) → o
Vector(Z_1, A_2) → u
Line(A_2, u) → d_2
Point(d_2) → E_2
Point(d_2) → F_2
Vector(E_2, F_2) → z
If(Tutorial ≟ true, If[Progress ≟ 0, true, false], false) → tutorial_{1st(2)}
If(Tutorial ≟ true, If[Progress ≟ 0 ∨ Progress ≟ 1, false, true], false) → tutorial_{1stto2nd}
If(Tutorial ≟ true, If[Progress ≟ 0, true, false], false) → tutorial_{1st}
If(Tutorial ≟ true, If[Progress ≟ 1 ∧ Progress_1 ≟ 0, true, false], false) → Tutorial_{2nd}
If(Tutorial ≟ true, If[Progress ≟ 1, false, true], false) → Vector_1
Angle(K'', J', L'') → ο
Angle(B', A', C') → ρ
Ray(G, J) → e_2
Ray(D, A) → f_2
OrthogonalLine(G_2, f_2) → g_2
Intersect(g_2, f_2) → H_2
Segment(G_2, J) → h_2
Segment(H_2, A) → i_2
Midpoint(G_2, H_2) → I_2
Vector(I_2, G_2) → z_1
Vector(I_2, H_2) → u_2
If(Progress_2 ≟ 1, true, false) → aplusb
Circle(L'', (0.5 * a)) → k_2
Point(k_2) → J_2
Segment(L'', J_2) → j_2
Rotate(J_2, ((90 * Progress_3))°, L'') → K_2
Circle(B', (0.5 * b)) → p_2
Point(p_2) → L_2
Rotate(L_2, (-(((90 * Progress_3))°)), B') → M_2
If(Progress_3 ≟ 0, false, true) → Stillaandb
If(Progress_3 ≟ 0 ∨ Progress_3 ≟ 1, false, true) → movingaandb
If(Progress_3 ≟ 0, false, true) → movingrightangles
Circle(L'', sqrt(a^(2) + ((0.5 * b))^(2))) → q_2
Circle(B', sqrt(b^(2) + ((0.5 * a))^(2))) → s_2
Point(q_2) → N_2
Point(s_2) → O_2
Rotate(N_2, ((90 * Progress_3))°, L'') → P_2
Rotate(O_2, (-(((90 * Progress_3))°)), B') → Q_2
Intersect(c_3, m_2) → R_2,
Rotate(R_2, ((90 * Progress_3))°, L'') → T_2
Intersect(d_3, n_2) → S_2,
Angle(J, L'', J') → σ
Angle(A', B', A) → τ
If(Progress_4 ≟ 1, true, false) → SecondBlueSquare
If(Progress_3 ≟ 1, true, false) → Progress4_{visibility}
If(Progress_2 ≟ 1, If[Progress_4 ≟ 0, true, false], false) → Progress3_{visibility}
If(Progress_4 ≟ 0, true, false) → Hypo_{1st}
If(Progress_4 ≟ 0, true, false) → Legs_{1st}
If(Progress_2 ≟ 1, If[Progress_3 ≟ 1, false, true], false) → fakecsquared
If(Progress_3 ≟ 1, If[Progress_4 ≟ 1, false, true], false) → SecondEquation
Vector(U_2, V_2) → v_2
If(Progress ≟ 0, false, true) → f
If(Tutorial ≟ true, If[Progress_1 ≟ 1, false, true], false) → AnnoyingTutorial

AI Drawing Prompt Reference

Draw a right triangle and demonstrate the Pythagorean Theorem by moving and rearranging its parts. Start with a shape composed of the triangle's sides, then rotate and translate a part of it to combine with another part, forming a new shape. Use circles and lines to illustrate area relationships. The dynamic process shows the transformation and overlap of shapes, proving the geometric meaning of the theorem.

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