User:Eznutella888
Hello fellow users of AOPS, my name is ! As you can see I like math. That's why I'm here.
I have taken many math competitions, including the Canadian Gauss, Pascal and Cayley. I have also taken Canadian Intermediate Mathematics Examination, and the Math Challengers competition sponsored by the Canadian Math Challengers Society. I also have taken AMC 8, and this year I'm taking the AMC 10, as well as the COMC (Canadian Open Mathematics Challenge).
We can set coordinates for the points. . The line
's equation is
,
's equation is
, and line
's equation is
. Adding the equations of lines
and
, the coordinates of
is
. Furthermore the coordinates
is
. Using the Pythagorean Theorem, the length of
is
, and the length of
=
PQ = DP - DQ = \frac{9\sqrt{5}}{5} - \frac{9\sqrt{5}}{4} = \frac{9\sqrt{5}}{20}. The length of
. Then
BP : PQ : QD: = frac{3\sqrt{5}}{4} : \frac{9\sqrt{5}}{20} : \frac{9\sqrt{5}}{5} = 15\sqrt[2]{5} : 9\sqrt[2]{5} : 36\sqrt[2]{5} = 15 : 9 : 36 = 5 : 3 : 12. Then
and
is
and
, respectively. The problem tells us to find
, so