Trigonometry or something else?

Geometry Level 4

The A-excircle of triangle $ABC$ is defined to be the unique circle which is tangent to lines $AB$ , $BC$ and $AC$ of triangle $ABC$ which lies on the opposite side of line $BC$ as $A$ .

Triangle $ABC$ has side lengths $BC = 98$ , $AC = 76$ , $AB = 54$ , and its A-excircle is tangent to line $AC$ at $D$ . Find the length of $CD$ up to three decimal places.

The answer is 38.00.

1 solution

Manuel Kahayon
May 9, 2017

We let the A-excircle meet $BC$ at $E$ , and $AB$ at $F$ . Notice that two tangents from a point to a circle are always equal, so $BE = BF$ , and $CE = CD$ .

Notice then that

$AB + BC + AC = AB + (BE + EC) + AC = (AB + BF) + (AC + CD) = AF + AD$ , giving us

$AB + BC + AC = AD + AF$ .

We then have that $98+76+54 = 228 = AD + AF$ . But since AD and AF are both tangents from $A$ to the A-excircle, we have $AD = AF$ . In particular, $228 = 2AD$ , giving us $AD = 114$ . Now, we can easily see that $CD = AD - AC = 114 - 76 = 38$ .

Thus, our final answer is $\boxed{38}$ .

Trigonometry or something else?

The answer is 38.00.

1 solution

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