Triangle 90<x<135 such that B is tangent

  • #1

Homework Statement


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?


Homework Equations


Well its obvious that it must be greater than 90 degrees due to cyclic quads.


The Attempt at a Solution


Ive attempted and physically shown its impossible to have one greater than 135 degrees, and ive actually proven the 90 degrees. How do i prove the 135 degree cap? And how do i do the other part?

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
ehild
Homework Helper
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Make a drawing and find relations among the angles.
You need to find out something about the angle gamma.

Hint: Find angle x in terms of gamma, then the green one in the yellow right triangle in terms of gamma and beta.
How is it related to the red angle ? What is the other angle in the blue triangle?

ehild
 

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  • #3
Could you please start me off as to finding X in terms of gamma, i realise that parallelograms have equal opposite angles....
 
  • #4
ehild
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There is no parallelogram in the picture. I draw a new picture There is a circle with central angle (red) and inscribed angle gamma (blue) belonging to the same intercepted arc. You certainly know how the central angle and inscribed angle are related. Can you find φ and then x?

ehild
 

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  • #5
Well ... must equal 120 degrees, and X=30 degrees.
 
  • #6
So that means in the original picture, gamma = 120 degrees as well.
 
  • #7
ehild
Homework Helper
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Why 120 degrees???

The blue angle (gamma) is an inscribed angle for the arc shown in the picture. The red angle is the central angle belonging to the same arc. The central angle is twice the inscribed angle, so it is 2γ. can you find out φ and x now?

ehild
 
  • #8
... is y and angle X is 1/2y
 
  • #9
ehild
Homework Helper
15,543
1,909
This is a complicated problem, and my English is rather poor. I try my best, but I am afraid, I can not explain the solution to you if you do not help. Have you learnt about angles in the circle? Central angle, inscribed angle? Do you understand why is the red angle 2γ?
You have to prove that the largest angle in the triangle ABC is smaller than 135°. I call that angle γ.

The red angle and φ are conjugate angles, sum to one turn, 360°. If the red one is 200°, for example, φ=360-200=160°. What is φ if the red angle is 2γ?

ehild
 

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