Trivial Geometry

Geometry Level 3

Let $I$ be the incenter of an acute $\triangle ABC$ . Let $A', B', C'$ be the reflections of $I$ in lines $BC, CA, AB$ respectively. Find $\dfrac{\angle B'IC'}{\angle B'A'C'}$ .

The answer is 2.

1 solution

TheKnee OfJustice
Mar 10, 2014

Nice problem. Basically, (not giving away the whole problem), you need to deduce that I is the circumcentre of A'B'C'. Then the problem is trivial.

Proof of the fact that $I$ is the circumcenter of $A'B'C'$ :

Note that $I$ is equidistant from $BC, CA, AB$ , and hence by virtue of reflection, $I$ is also equidistant from $A', B', C'$ , which implies that $I$ is the circumcenter of $\triangle A'B'C'$ .

I wanted to make a problem which uses this fact as a lemma. Unfortunately, I couldn't come up with anything else. :(

Sreejato Bhattacharya - 7 years, 3 months ago

Trivial Geometry

The answer is 2.

1 solution

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