Can you help me to solve this problem ?
The answer must be just by geometric prove .Not by trigonometric !!!
From the figure, we know that
So,
From triangle ACD, we know that
So, we get some equations now.
Let's give some names to the other angles:
From triangle ABD, we know that:
From triangle DBC, we know that:
And lastly, we know that
and
There is enough information to find alpha
Unfortunately, that is not enough information to find alpha. There are four equations for four unknowns, but they are not independent. (The fourth equation, in its corrected form above, is just what you get by adding the first two equations and subtracting the third.)
By trigonometry I get the answer to be that , but I cannot see any way of proving that by synthetic geometry. This seems to be a variant of the notorious 80-80-20 triangle problem, but maybe this one is even harder.