In ΔABC, r_1 , r_2 and r_3 are the exradii
as shown. Prove that r_1 = (Δ/(s − a)) ,
r_2 = (Δ/(s − b)) and r_3 = (Δ/(s − c)) . Here
s = ((a + b + c)/2) .
let a_1 >a_2 >0 and a_(n+1) =(√(a_n a_(n−1 ) ))
where n is greater than equal to 2
Then
The sequence {a_(2n) } is
(1) monotonic increasing
(2)monotonic decreasing
(3)non monotonic
(4)unbounded
Let K, L, M and N be the midpoints of
the sides AB, BC, CD and DA,
respectively, of a cyclic quadrilateral
ABCD. Prove that the orthocenters
of the triangles AKN, BKL, CLM and
DMN are the vertices of a
parallelogram.
Let ABCD be a convex quadrilateral.
Prove that the orthocenters of the
triangles ABC, BCD, CDA and DAB
are the vertices of a quadrilateral
congruent to ABCD and prove that the
centroids of the same triangles are the
vertices of a cyclic quadrilateral.
Let A′, B′ and C′ be points on the sides
BC, CA and AB of the triangle ABC.
Prove that the circumcircles of the
triangles AB′C′, BA′C′ and CA′B′
have a common point. Prove that the
property holds even if the points A′,
B′ and C′ are collinear.
In the interior of a quadrilateral
ABCD, consider a variable point P.
Prove that if the sum of distances from
P to the sides is constant, then ABCD
is a parallelogram.
Let ABCD be a convex quadrilateral
and let E and F be the points of
intersections of the lines AB, CD and
AD, BC, respectively. Prove that the
midpoints of the segments AC, BD,
and EF are collinear.
Let ABCD be a convex quadrilateral
and M a point in its interior such that
[MAB] = [MBC] = [MCD] = [MDA].
Prove that one of the diagonals of
ABCD passes through the midpoint of
the other diagonal.