randRange(1, 10) HEIGHT_A + randRange(1, 20) randRange(1, 10) SIDE_B * HEIGHT_A / (HEIGHT_B - HEIGHT_A) ["x", HEIGHT_A, SIDE_B, HEIGHT_B] atan(HEIGHT_A / SIDE_A) * 180 / PI new Triangle([0, 0], [ANGLE, 90, 90 - ANGLE], 50 * (SIDE_A / (SIDE_A + SIDE_B)) * HEIGHT_A / HEIGHT_B, {}) function(){var e=new Triangle([0,0],[ANGLE,90,90-ANGLE],50,{});return e.labels={points:["A","D","E"]},e}()
SIDE_A

\overline{AB} : \overline{BC} = \dfrac{x}{HEIGHT_A}

\overline{AD} : \overline{DE} = \dfrac{x + SIDE_B}{HEIGHT_B}

\dfrac{x}{HEIGHT_A} = \dfrac{x + SIDE_B}{HEIGHT_B}

HEIGHT_Bx = HEIGHT_A(x + SIDE_B) x + SIDE_B

HEIGHT_Bx = HEIGHT_Ax + HEIGHT_A * SIDE_B

plus(HEIGHT_B - HEIGHT_A + "x") = HEIGHT_A * SIDE_B

x = fractionReduce(HEIGHT_A * SIDE_B, HEIGHT_B - HEIGHT_A)

randRange(1, 10) randRange(1, 10) randRange(1, SIDE_A + SIDE_B) HEIGHT_B * SIDE_A / (SIDE_A + SIDE_B) [SIDE_A, "x", SIDE_B, HEIGHT_B] atan(HEIGHT_A / SIDE_A) * 180 / PI new Triangle([0, 0], [ANGLE, 90, 90 - ANGLE], 50 * (SIDE_A / (SIDE_A + SIDE_B)) * HEIGHT_A / HEIGHT_B, {}) function(){var e=new Triangle([0,0],[ANGLE,90,90-ANGLE],50,{});return e.labels={points:["A","D","E"]},e}()
HEIGHT_A

\overline{AB} : \overline{BC} = \dfrac{SIDE_A}{x}

\overline{AD} : \overline{DE} = \dfrac{SIDE_A + SIDE_B}{HEIGHT_B} = fraction(SIDE_A + SIDE_B, HEIGHT_B)

\dfrac{SIDE_A}{x} = \dfrac{SIDE_A + SIDE_B}{HEIGHT_B}

SIDE_A * HEIGHT_B = SIDE_A + SIDE_Bx

x = fractionReduce(SIDE_A * HEIGHT_B, SIDE_A + SIDE_B)

randRange(1, 10) randRange(1, SIDE_A) HEIGHT_A + randRange(1, 10) SIDE_A * (HEIGHT_B - HEIGHT_A) / HEIGHT_A [SIDE_A, HEIGHT_A, "x", HEIGHT_B] atan(HEIGHT_A / SIDE_A) * 180 / PI new Triangle([0, 0], [ANGLE, 90, 90 - ANGLE], 50 * (SIDE_A / (SIDE_A + SIDE_B)) * HEIGHT_A / HEIGHT_B, {}) function(){var e=new Triangle([0,0],[ANGLE,90,90-ANGLE],50,{});return e.labels={points:["A","D","E"]},e}()
SIDE_B

\overline{AB} : \overline{BC} = fraction(SIDE_A, HEIGHT_A)

\overline{AD} : \overline{DE} = \dfrac{SIDE_A + x}{HEIGHT_B}

\dfrac{SIDE_A + x}{HEIGHT_B} = fraction(SIDE_A, HEIGHT_A)

HEIGHT_A(SIDE_A + x) SIDE_A + x = HEIGHT_B * SIDE_A

HEIGHT_A * SIDE_A + HEIGHT_Ax = HEIGHT_B * SIDE_A

plus(HEIGHT_A + "x") = HEIGHT_B * SIDE_A - HEIGHT_A * SIDE_A

x = fractionReduce(HEIGHT_B * SIDE_A - HEIGHT_A * SIDE_A, HEIGHT_A)

randRange(1, 10) randRange(1, 10) randRange(1, SIDE_A) (SIDE_A + SIDE_B) * HEIGHT_A / SIDE_A [SIDE_A, HEIGHT_A, SIDE_B, "x"] atan(HEIGHT_A / SIDE_A) * 180 / PI new Triangle([0, 0], [ANGLE, 90, 90 - ANGLE], 50 * (SIDE_A / (SIDE_A + SIDE_B)) * HEIGHT_A / HEIGHT_B, {}) function(){var e=new Triangle([0,0],[ANGLE,90,90-ANGLE],50,{});return e.labels={points:["A","D","E"]},e}()
HEIGHT_B

\overline{AB} : \overline{BC} = fraction(SIDE_A, HEIGHT_A)

\overline{AD} : \overline{DE} = \dfrac{SIDE_A + SIDE_B}{x} = \dfrac{SIDE_A + SIDE_B}{x}

fraction(SIDE_A, HEIGHT_A) = \dfrac{SIDE_A + SIDE_B}{x}

plus(SIDE_A + "x") = HEIGHT_A * (SIDE_A + SIDE_B)

x = fraction(HEIGHT_A * (SIDE_A + SIDE_B), SIDE_A) = fractionReduce(HEIGHT_A * (SIDE_A + SIDE_B), SIDE_A)

¿Cuál es el valor de x?

var tri_range=TRI_B.boundingRange(1);init({range:[[tri_range[0][0],tri_range[0][1]],[tri_range[1][0],tri_range[1][1]]],scale:400/(tri_range[0][1]-tri_range[0][0])}),style({strokeWidth:1,stroke:KhanUtil.BLACK});var x_max=TRI_B.boundingRange()[0][1],y_max=TRI_B.boundingRange()[1][1],line_x=x_max*SIDE_A/(SIDE_A+SIDE_B),line_y=y_max*HEIGHT_A/HEIGHT_B,square=(tri_range[0][1]-tri_range[0][0])/36;rect(x_max-square,0,square,square),rect(line_x-square,0,square,square),style({strokeWidth:2,stroke:KhanUtil.BLUE}),TRI_A.draw(),TRI_B.draw(),TRI_B.drawLabels(),style({color:"black"}),label([line_x,0],"B","below"),label([line_x,line_y],"C","above"),label([line_x/2,0],LABELS[0],"below"),label([line_x,line_y/2],LABELS[1],"right"),label([(SIDE_A+SIDE_B/2)*x_max/(SIDE_A+SIDE_B),0],LABELS[2],"below"),label([x_max,y_max/2],LABELS[3],"right")

\triangle ABC y \triangle ADE tienen ambos un ángulo recto y comparten \angle BAC.

Por lo tanto \triangle ABC y \triangle ADE son triángulos semejantes.

Por lo tanto, la razón \overline{AB} : \overline{BC} es igual a la razón \overline{AD} : \overline{DE}.