Construct a triangle, given its angle at the vertex, and the altitude and the median drawn to the base.

Question

The problem is from a book. Also it gives a hint: "Double the median extending it past the base, connect the endpoint the with the vertices at the base, and consider the parallelogram thus formed.".

The image below is my interpretation of the problem. In the left is some random triangle with the colored segments and angle as the given. In the right is some attempt at reconstructing it with the given (starting with the point A). I can't seem to reconcile the hint with my attempt: I can't construct the parellelogram that it is talking about. Maybe I'm misunderstanding the problem? Any hint?

score 4 · Answer 1 · answered May 31 '17 at 21:41

4

The black angle at the start is the angle that is given. The line is the base and the two red dots are the vertex and the median.

answered May 31 '17 at 21:41

Asinomás

105,651

Thank you sir. After studying why your solution works, I finally figure out how the hint can be used. – tighten Jun 01 '17 at 13:40
Yeah, the trick is to build a paralelogram with appropriate angles and two vertices on the line and use the fact that the diagonals bisect themselves – Asinomás Jun 01 '17 at 13:47
I used a different approach that the book uses a lot. Base on secant(chord) and tangent of a point on a circle (dunno exactly the name of the theorem). – tighten Jun 01 '17 at 13:52
http://imgur.com/a/OTMsJ – tighten Jun 01 '17 at 13:53
its basically the same, you just transpassed the angle smarter – Asinomás Jun 01 '17 at 14:04

Construct a triangle, given its angle at the vertex, and the altitude and the median drawn to the base.

1 Answers1