I have to proof in a triangle, that $\alpha_1>\alpha_2$ holds. The inner point P (from where I draw the smaller triangle) is set randomly.
Does anyone have a suggestion where I have to start?
Greetings
I have to proof in a triangle, that $\alpha_1>\alpha_2$ holds. The inner point P (from where I draw the smaller triangle) is set randomly.
Does anyone have a suggestion where I have to start?
Greetings
Given that the origin triangle is $\triangle ABC$, and the inner point is $P$, where $\alpha_1 = \angle BPC$, $\alpha_2 = \angle BAC$. Supposing that $D$ is the intersection of $AP$ and $BC$, we have $\angle BPC = \angle BPD + \angle DPC = \angle BAP + \angle ABP + \angle CAP + \angle ACP$ $= \angle BAC + \angle ABP + \angle ACP > \angle BAC$, Q.E.D.
Each of $\,\alpha_1\,,\,\alpha_2\,$ equals $\,180 - (\beta_1+\beta_2)\,$ , with $\beta_i$ being the other two angles in the big (in the little red) triangle. As the other two angles of $\,\alpha_1\,$ are each less than the other two angles of $\,\alpha_2\,$ we get what we want.
HINT: Draw a line that passes through the vertices of the two angles and the opposite side.