Question

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

Answer #1

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.

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.

a) In the triangle ABC, angle A is 60 ° and angle B 90 °. The
side AC is 100 cm. How long is the side BC? Determine an exact
value.
b) An equilateral triangle has the height of 11.25 cm. Calculate
its area.

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

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?

5. Suppose that the incenter I of ABC is on the triangle’s Euler
line. Show that the triangle is isosceles.
6. Suppose that three circles of equal radius pass through a
common point P, and denote by A, B, and C the three other points
where some two of these circles cross. Show that the unique circle
passing through A, B, and C has the same radius as the original
three circles.
7. Suppose A, B, and C are distinct...

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

Three parallel lines are such that one passes through each
vertex of a triangle ABC, and they are not parallel to any of the
triangles sides. The line through A meets BC (extended if
necessary) in X, the lines through B and C meet CA and AB in Y and
Z respectively. Prove that area(XYZ) = 2xArea(ABC)

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