MathDB

13

Part of 2019 BMT Spring

Problems(2)

area of quadr. wanted, 13-14-15 triangle, incenter excenter 2019 BMT Team 13

Source:

1/5/2022
Triangle △ABC\vartriangle ABC has AB=13AB = 13, BC=14BC = 14, and CA=15CA = 15. △ABC\vartriangle ABC has incircle γ\gamma and circumcircle ω\omega. γ\gamma has center at II. Line AIAI is extended to hit ω\omega at PP. What is the area of quadrilateral ABPCABPC?
geometryincentercircumcircle
(CADB)=, 2 intersecting circles, AC_|_AB, BD_|_AB 2019 BMT Individual 13

Source:

1/5/2022
Two circles O1O_1 and O2O_2 intersect at points AA and BB. Lines AC‾\overline{AC} and BD‾\overline{BD} are drawn such that CC is on O1O_1 and DD is on O2O_2 and AC‾⊥AB‾\overline{AC} \perp \overline{AB} and BD‾⊥AB‾\overline{BD} \perp \overline{AB}. If minor arc AB=45AB= 45 degrees relative to O1O_1 and minor arc AB=60AB= 60 degrees relative to O2O_2 and the radius of O2=10O_2 = 10, the area of quadrilateral CADBCADB can be expressed in simplest form as a+bk+cℓa + b\sqrt{k} + c\sqrt{\ell}. Compute a+b+c+k+ℓa + b + c + k +\ell.
geometrycircles