Question

If H is the orthocenter of the triangle ABC and AH intersects BC at D and circumcircle of triangle ABC at E, find HD:DE.

Answer #1

HD:DE=1:1

Triangle ABC is a right angle triangle in which
∠B = 90 degree, AB = 5 units , BC = 12 units. CD
and AE are the angle bisectors of ∠C and ∠A
respectively which intersects each other at point I. Find the area
of the triangle DIE.

In an isosceles triangle ABC ,AB=BC,angle B=20 . M and N are on
AB and BC respectively such that angle MCA =60, angle NAC =50.find
angle MNC

ABC is a right-angled triangle with right angle at A, and AB
> AC. Let D be the midpoint of the side BC, and let L be the
bisector of the right angle at A. Draw a perpendicular line to BC
at D, which meets the line L at point E. Prove that
(a) AD=DE; and
(b) ∠DAE=1/2(∠C−∠B)
Hint: Draw a line from A perpendicular to BC, which meets BC in
the point F

Suppose that the incircle of triangle ABC touches AB at Z, BC at
X, and AC at Y . Show that AX, BY , and CZ are concurrent.

Let J be a point in the interior of triangle ABC. Let D, E, F be
the feet of the perpendiculars from J to BC, CA, and AB,
respectively. If each of the three quadrilaterals AEJF, BFJD, CDJE
has an inscribed circle tangent to all four sides, then J is the
incenter of ∆ABC. It is sufficient to show that J lies on one of
the angle bisectors.

Consider the triangle ABC. Suppose that the perpendicular
bisectors of line segments AB and BC intersect at point X. Prove
that X is on the perpendicular bisector of line segment AC.

Let O be the center of circumscribed circle of ABC triangle. Let
a, b, c be the vectors pointing from O to the vertexes. Let M be
the endpoint of a + b + c measured from O. Prove that M is the
orthocenter of ABC triangle.

If in triangle ABC and Triangle XYZ we have AB = XY, AC = XZ,
but m<A > m<X, then BC > YZ. Conversely, if BC > YZ
then m<A > m<X.

NON EUCLIDEAN GEOMETRY
Prove the following:
Claim: Let AD the altitude of a triangle
▵ABC. If BC is longer than or equal to AB and AC, then
D is the interior of BC.
What happens if BC is not the longest side? Is D still always in
the interior of BC? When is D in the interior?

Suppose in triangle ABC ma and mb are the
lengthsof the medians from A and B respectively. Prove that if a≥b
then ma≤mb (recall the convention that in
triangle ABC, a=BC and b=AC

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