The Feet Of Perpendiculars From The Internal Bisector

Geometry Level 3

In acute triangle $\triangle ABC,$ the internal angle bisector of $\angle BAC$ meets $BC$ at $D.$ $E, F$ are the feet of perpendiculars from $D$ on $AC, AB$ respectively. Let $O$ be the circumcenter of $\triangle AEF.$ Given that $\angle BAC = 60^{\circ},$ find $\angle BAO$ in degrees.

Details and assumptions

The image shown is not accurate.

The answer is 30.

2 solutions

Sreejato Bhattacharya
Apr 6, 2014

Image link: http://s30.postimg.org/i2a71eea9/Untitled.png

Since $\angle DFA = \angle DEA= 90^{\circ},$ the circumcircle of $\triangle AEF$ passes through $D$ and $AD$ is one of its diameters. It follows that $O$ is the midpoint of $AD.$ Our desired answer is $\angle BAO = \angle BAD= \dfrac{\angle BAC}{2} = \boxed{30^{\circ}}.$

<1100 rating problem :3

Sagnik Saha - 7 years, 2 months ago

I know, this problem was meant to be trivial.

Sreejato Bhattacharya - 7 years, 2 months ago

Cheeky little photo! :)

David Kroell - 7 years, 1 month ago

Instead of providing an inaccurate figure you should have not marked the center.Anyway a very simple question .

Pushpak Roy - 7 years, 1 month ago

Nice question, good solution

Milun Moghe - 7 years, 2 months ago

easy question but nice.

BHANU VISHWAKARMA - 7 years, 2 months ago

O lies on AD, that's all.

Richard Rodriguez - 7 years, 1 month ago

This problem sucks!The photo was incorrect!

Alrazi Turjo - 7 years, 1 month ago

When you are making a Geometry problem, don´t judge by the diagram, you could guess other things that are necessarily correct. For example, everyone would at first think that the center is not on the bisecter, but before making any conclusions one should see other facts and see if the center is on the bisecter or not. There are even problems where you prove that some points are the same and transform the problem in another. I also thought that, I was looking a way to prove it, and found out that the wanted angle measured 30°, so at the end I knew that the center O was on AD, so try making hypothesis and prove them... but don´t negate a possibility that could occur.

Fernando Raúl Cortez Chávez - 7 years, 1 month ago

I did mention that the diagram is incorrect.

Sreejato Bhattacharya - 7 years, 1 month ago

Too easy !!!

Arghyanil Dey - 7 years, 1 month ago

Ashish Sharma
Apr 17, 2014

The angles at circumference (AFD and AED) are given to be 90 degrees each. So the point O must lie in AD which eventually turns out to be a diameter. With that invented, angle BAO will be the half of the given angle FAE, that is 30 degrees.

The Feet Of Perpendiculars From The Internal Bisector

The answer is 30.

2 solutions

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