Track / Stage 4 / 197 of 340 #457 of 1964

Problem 457

AMC 12 late, AIME early

Geometry Difficulty 4.8 Find the answer

3. Given a right trapezoid $A B C D$ with side lengths $A B=2, B C=C D=10, A D=6$ , a circle is drawn through points $B$ and $D$ , intersecting the extension of $B A$ at point $E$ and the extension of $C B$ at point $F$ . Then the value of $B E-B F$ is $\qquad$

Your answer

A number or a short expression. Spacing, $ signs and \frac vs / are all fine.

Official solution

3.4 .

As shown in Figure 3, extend $C D$ to intersect $\odot O$ at point $G$ . Let the midpoints of $B E$ and $D G$ be $M$ and $N$ , respectively. It is easy to see that $A M = D N$ . Since $B C = C D = 10$ , by the secant theorem, it is easy to prove that
$\begin{aligned} B F = & D G. \text{ Therefore, } \\ & B E - B F = B E - D G = 2(B M - D N) \\ & = 2(B M - A M) = 2 A B = 4 . \end{aligned}$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.