Track / Stage 4 / 282 of 340 #542 of 1964

Problem 542

AMC 12 late, AIME early

Geometry Difficulty 4.9 Find the answer

Through the vertex of isosceles $\triangle A B C$ , draw a line intersecting the extension of the opposite side at $D$ . If the resulting triangles are all isosceles, then $\triangle A B C$ has $\qquad$ such configurations.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

4.5.

Draw a straight line from the base angle to intersect with the extension of the opposite side, there are 5 $\triangle A B C$ that meet the conditions, with base angles of $45^{\circ} 、 72^{\circ} 、 36^{\circ} 、 \frac{180^{\circ}}{7}$ 、 $\frac{2 \times 180^{\circ}}{7}$ .
Draw a straight line from the vertex angle to intersect with the extension of the opposite side, there is 1 $\triangle A B C^{\prime}$ that meets the conditions, with a base angle of $72^{\circ}$ , which is repeated above.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.