One angle of the triangle must be..

Question

In a triangle ABC the altitudes from B and C on the opposite sides are not shorter than their respective opposite sides. then one of the angle of the triangle ABC is?

Answer 1

In a triangle, an altitude is not greater than any of the two sides with the same vertex. In our case, if $BH$ and $CK$ are the altitudes, we have then: $$ BH\le AB \quad\hbox{and}\quad CK\le AC. $$ On the other hand we know that $$ BH\ge AC \quad\hbox{and}\quad CK\ge AB. $$ Combining the above inequalities we get: $$ AB\ge AC \quad\hbox{and}\quad AB\le AC, \quad\hbox{whence:}\quad AB=AC. $$ Thus $ABC$ is an isosceles triangle, which entails $BH=CK$. We have then: $$ BH\le AB\le CK = BH, \quad\hbox{whence:}\quad BH=AB. $$ We come then to the conclusion that in our triangle altitudes $BH$ and $CK$ are the same as sides $AB$ and $AC$, which can happen only if $\angle BAC=90°$.