Proving $\angle ABC$ is Acute: Inside the Triangle $ABC$

[maxkor]

Inside the triangle $ABC$ is point $P$, such that $BP > AP$ and $BP > CP$. Prove that $\angle ABC$ is acute.

 

[mrtwhs]

We are given that $x>y$ and $w>v$ therefore $x+w > y+v = \angle ABC$

Now $\angle CAB > x$ and $\angle ACB >w$ so $\angle CAB + \angle ACB > x+w > y+v = \angle ABC$

But $\angle CAB + \angle ACB = 180 - \angle ABC$ so $180 - \angle ABC > \angle ABC$ and $\angle ABC < 90$

 

[topsquark]

I don't get it. Why doesn't the triangle below fit the ciiteria?

-Dan

 

[maxkor]

@topsquark
Why doesn't the triangle fit the criteria?
This triangle fits the criteria.

 

[topsquark]

@topsquark
Why doesn't the triangle fit the criteria?
This triangle fits the criteria.

Aaahh! I was reading the problem wrong. I thought it was asking to show that the triangle ABC was acute. My bad.

-Dan