1954 AHSME Problems/Problem 19

Problem 19

If the three points of contact of a circle inscribed in a triangle are joined, the angles of the resulting triangle:

$\textbf{(A)}\ \text{are always equal to }60^\circ\\ \textbf{(B)}\ \text{are always one obtuse angle and two unequal acute angles}\\ \textbf{(C)}\ \text{are always one obtuse angle and two equal acute angles}\\ \textbf{(D)}\ \text{are always acute angles}\\ \textbf{(E)}\ \text{are always unequal to each other}$

Solution

For the sake of clarity, let the outermost triangle be $ABC$ with incircle tangency points $D$ , $E$ , and $F$ on $BC$ , $AC$ and $AB$ respectively. Let $\angle A=\alpha$ , and $\angle B=\beta$ . Because $\triangle AFE$ and $\triangle BDF$ are isosceles, $\angle AFE=\frac{180-\alpha}{2}$ and $\angle BFD=\frac{180-\beta}{2}$ . So $\angle DFE=180-\frac{180-\alpha}{2}-\frac{180-\beta}{2}=\frac{\alpha+\beta}{2}$ , and since $\alpha + \beta <180^\circ$ , $\angle DFE$ is acute.

The same method applies to $\angle FED$ and $\angle FDE$ , which means $\triangle DEF$ is acute - hence our answer is $\fbox{D}$ .

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1954 AHSME Problems/Problem 19

Problem 19

Solution