Thread: Proving an angle between 90 and 135 degrees given properties.

In triangle ABC, the altitude from B is tangent to the circumcircle of ABC. Prove that the largest angle of the triangle is between 90◦ and 135◦. If the altitudes from both B and from C are tangent to the circumcircle, then what are the angles of the triangle?

I have made a start and drawn a figure although am unsure which method to use in actually proving that this always occurs.
Any help would be great
Cheers

Originally Posted by pikachu26134

In triangle ABC, the altitude from B is tangent to the circumcircle of ABC. Prove that the largest angle of the triangle is between 90◦ and 135◦. If the altitudes from both B and from C are tangent to the circumcircle, then what are the angles of the triangle?

I have made a start and drawn a figure although am unsure which method to use in actually proving that this always occurs.
Any help would be great
Cheers

If the largest angle is less than 90 then all the angles are less than 90, but then the triangle becomes acute angled and hence the altitude from B will lie inside the triangle and hence INSIDE the circumcircle of the triangle and as a result the altitude can't be tangent to the circumcircle. So we conclude that the largest angle is greater than 90.
now if you take the largest angle to be equal to 135 then you get a degenerate triangle(prove it).
if you take the largest angle to be greater than 135 then you don't get a triagle at all.

How do i go about proving 90-135 is degenerate?? I don't have a clue.

Originally Posted by pikachu26134

How do i go about proving 90-135 is degenerate?? I don't have a clue.

NO NO, 90 to 135 is not degenerate. When the largest angle is equal to 135 then the triangle is degenerate. If the largest angle is between 90 and 135 then you get an obtuse angled triangle.