MathDB

3

Part of 2006 Spain Mathematical Olympiad

Problems(2)

distances of a center in circle from sides of isosceles triangle

Source: 2006 Spanish Mathematical Olympiad P3

7/20/2018
ABCABC is an isosceles triangle with AB=ACAB = AC. Let PP be any point of a circle tangent to the sides ABAB in BB and to AC in C. Denote aa, bb and cc to the distances from PP to the sides BC,ACBC, AC and ABAB respectively. Prove that: a2=bca^2=bc
geometrycircumcircledistanceisosceles
\sqrt{S_1}+\sqrt{S_2}\le \sqrt{S} in convex ABCD

Source: 2006 Spanish Mathematical Olympiad P6

7/20/2018
The diagonals ACAC and BDBD of a convex quadrilateral ABCDABCD intersect at EE. Denotes by S1,S2S_1,S_2 and SS the areas of the triangles ABEABE, CDECDE and the quadrilateral ABCDABCD respectively. Prove that S1+S2≤S\sqrt{S_1}+\sqrt{S_2}\le \sqrt{S} . When equality is reached?
geometrygeometric inequalityconvex quadrilateraldiagonalssquare rootsareas