Question

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

Answer #1

Given △ABC, extend sides AB and AC to rays AB and AC forming
exterior angles. Let the line rA be the angle bisector ∠BAC, let
line rB be the angle bisector of the exterior angle at B, and let
line rC be the angle bisector of the exterior angle at C.
• Prove that these three rays are concurrent; that is, that they
intersect at a single point. Call this point EA
• Prove that EA is the center of...

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.

Consider a quadrilateral ABCD such that ∠BAD and ∠ADC are
perpendicular, the rays AB and CD are on the same side of the line
AD, and AB ≅ CD.
Prove the following claims on E2, H2, and
S2.
∠ABC ≅ ∠DCB.
the perpendicular bisector of AD is also the perpendicular
bisector of BC.
Hint: Look for symmetries.

In triangle ABC , let the bisectors of angle b meet AC at D and
let the bisect of angle C meet at AB at E. Show that if BD is
congruent to CE then angle B is congruent to angle C.

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.

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?

Consider a quadrilateral ABCD such that ∠BAD and ∠ADC are
perpendicular, the rays AB and CD are on the same side of the line
AD, and AB ≅ CD. Quadrilaterals with these properties are called
Khayyam quadrilaterals
Prove the following claims on E2, H2, and
S2.
∠ABC ≅ ∠DCB.
the perpendicular bisector of AD is also the perpendicular
bisector of BC.
Hint: Look for symmetries.

In ⊿???, ∠? is a right angle. D is the midpoint of AC, F is the
midpoint of BC and E is the midpoint of CF. AB =12 cm. BC = 16 cm.
What is the number of square centimeters of ⊿???? Describe and
justify your steps

Let ABCD be a rectangle with AB = 4 and BC = 1. Denote by M the
midpoint of line segment AD and by P the leg of the perpendicular
from B onto CM.
a) Find the lengths of P B and PM.
b) Find the area of ABPM.
c) Consider now ABCD being a parallelogram. Denote by M the
midpoint of side AD and by P the leg of the perpendicular from B
onto CM. Prove that AP =...

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...

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