I know that we are not supposed to discuss Brilliant problem answers, but this post is concerning a problem that was posted when Brilliant first started, and it doesn't even have a rating or a discussion, so if you get it wrong (like me) you can't discuss the answer. Anyways, some of you might have seen the article on Business Insider, the one about the 10 "smartest" kids in the world. If you haven't, that 's okay. But one of the problems is super hard. I've been working on it for months, and I even drew a 4x5 foot diagram of it on my wall in my room. Here's the problem:

Triangle ABC has AC = BC, angle ACB is 96 degrees. D is a point in ABC such that angle DAB is 18 degrees, and angle DBA is 30 degrees. What is the measure (in degrees) of angle ACD?

If any of you can post a good solution to this problem, I will respect you forever. I think that the answer is 42, but I could be wrong. I mean, the answer to everything is 42 (reference to HGTG)! Thank you so much!

## Comments

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TopNewestThe answer is \(78^\circ\), not \(80^\circ\). In a problem like this, one approach is to construct the point another way, and then show that it has the same defining properties as the point given in the problem.

Let \(E\) be the point inside triangle \(ABC\) such that \(AE = AC\) and \(\angle CAE = 24^\circ\). Let \(F\) be the point inside triangle \(ACE\) such that triangle \(CEF\) is equilateral.

http://i.imgur.com/Fb0iir4.png

What can you say about triangles \(AFC\) and \(BEC\)? What does that say about points \(D\) and \(E\)? – Jon Haussmann · 2 years, 9 months ago

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– Finn Hulse · 2 years, 9 months ago

Brilliant! and then the angle of the bases of the isosceles triangles are both 78! Amazing!Log in to reply

I proved it. – Ashu Pateriya · 1 year, 3 months ago

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There is one more (more geometric, no trigonometry) solution. Let us costruct the rombus ACBE. The angle AEB will also be 96 degrees. As the sum of the angles DAB and ABD is 48 degrees, then the point D will occur on the circumference with the center E. And AE=ED. The angle DAE equals 60 degrees, then EA=ED=AD. And AC=AD and we come to the same result - ACD equals 78 degrees. – Сергей Кротов · 2 years, 9 months ago

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Dear Finn Hulse. The answer is 78 degrees. To get this you should prove that AD=AC. In my solution I considered two triangles ACK (K lies in the center of AB) and ADB. Then AK=AB/2=ACsin48 and AD/sin30=AB/sin48 (a sine-theorem). You then directly get AC=AD. – Сергей Кротов · 2 years, 9 months ago

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– Finn Hulse · 2 years, 9 months ago

Don't worry about being too complex. I'm in calculus.Log in to reply

dear Mr. Finn Hulse actually there are missing data in the problem. i tried to solve it many times but always the problem redduces to to the following simultaneous equations x+y = 228

y+z = 168 x- z = 60 At the end of the day , if you can get a solution to these equations and find the value of z , your required angle is ( 96 - z ) . . . . .now I think the problem is how to find z ? thanks yours : aziz alasha – Aziz Alasha · 2 years, 9 months ago

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– Finn Hulse · 2 years, 9 months ago

Same here! Apparently you have to start drawing lines to solve it.Log in to reply

http://math.stackexchange.com/questions/245608/find-an-angle-in-a-given-triangle – Shamik Banerjee · 2 years, 9 months ago

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If you can make it on the wall, you can also make it on paper with protractor and scale and can landed on right answer. That's what i did and the correct answer is 80 degrees. – Raghav Garg · 2 years, 9 months ago

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– Conor Sokolowsky · 2 years, 9 months ago

He doesn't have a protractor. But can you prove it?Log in to reply

– Finn Hulse · 2 years, 9 months ago

Conor is correct. I don't like to use protractors to solve math. But thank you so much for taking the time to solve this! It's very much appreciated. Now all I need to know is why.Log in to reply