How large is angle CKA?
Triangle ABC has angles A=72°, B=60°, C=48°.
A circle is inscribed in a triangle ABC. The circle touches AB and BC at points E and F, respectively. Angle bisector A intersects line EF at point K.
- DukeLv 71 decade agoFavorite Answer
The answer is angle(CKA) = 90°.
Let I is the incenter of ∆ABC (intersection point of angle bisectors of A and B). Since both tangents from B to the incircle are equal, ∆BEF is equilateral hence
angle(AEF) = angle(AEK) = 120°, angle(BAK) = 36°, angle(AKE) = 24°
Consider ∆AEK and ∆AIC:
angle(EAK) = angle(IAC) = 36°, angle(AKE) = angle(ACI) = 24°,
angle(AEK) = angle(AIC) = 120°, i.e. they are similar, the coefficient of similarity is
| AE | : | AI | = cos 36° (∆AEI is right)
Then also | AK | : | AC | = cos 36°, but the latter is possible only if ∆AKC is right.
By the way K is outside ∆ABC because the line AK intersects BC in a point let's say A' such that ∆ABA' has angles 36°, 60°, 84° and IF as inradius is perpendicular to BC.
- sahsjingLv 71 decade ago
Do you have a picture for the problem? It's hard for me to know whether K is inside or outside of the circle.
I just figured out that K must be outside the circle.
Duke got it done before me.
Here is my approach:
Let O be the incenter, and T be the intersection point of the angle bisector and BC.
First, we can prove that
∆COT ~ ∆KFT because angle OCT = 24 degrees, and angle FKT = 180-36-120 = 24 degrees, and angle CTO and angle KTF are vertical angles.
OT:FT = CO:KF
Since angle CTK and angle OTF are vertical angles, they are congruent.
So, we have
∆CTK ~ ∆OTF (by SAS ~ theorem)
angle CKT = angle OFT = 90 degrees
angle CKA = angle CKT = 90 degrees