# All Triangles Are Isosceles

Geometry Level 4

Let $$\Delta ABC$$ be an arbitrary triangle.

Point $$P$$ is the midpoint of line $$BC$$, and line $$PQ$$ is perpendicular to line $$BC$$.

Line $$AQ$$ bisects angle $$\angle BAC$$.

Lines $$AB$$ and $$AC$$ are extended as necessary to include points $$R$$ and $$S$$ such that $$\Delta AQR$$ and $$\Delta AQS$$ are right triangles.

By reason of symmetry, line segments $$BQ$$ and $$CQ$$ are equal.

By reason of symmetry, line segments $$RQ$$ and $$SQ$$ are equal.

Therefore, by reason of congruent right triangles, line segments $$BR$$ and $$CS$$ are equal.

But, also by reason of congruent right triangles, line segments $$AR$$ and $$AS$$ are also equal.

Therefore, line segments $$AB$$ and $$AC$$ are equal, and triangle $$\Delta ABC$$ is isosceles.

Which of the following multiple choice answers is correct?

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