Question

Let
triangle ABC be a triangle with excenters Ia, Ib, and Ic. Assume
that triangle IaIbIc is equilateral and show triangle ABC is
equilateral.

Answer #1

Prove the following equation
Ia 1 1 1 Ia0
Ib = 1 a2 a Ia1
Ic 1 a a2 Ia2

Let A be an event, and let IA be the associated indicator random
variable: IA(ω)=1 if ω∈A, and IA(ω)=0 if ω∉A. Similarly, let IB be
the indicator of another event, B. Suppose that, P(A)=p, P(B)=q,
and P(A∪B)=r.
Find E[(IA−IB)2] in terms of p,q,r?
2.Determine Var(IA−IB) in terms of p,q,r?

Let A be an event, and let IA be the associated indicator random
variable (IA is 1 if A occurs, and zero if A does not occur).
Similarly, let IB be the indicator of another event, B. Suppose
that P(A)=p, P(B)=q, and P(A∩B)=r.
Find the variance of IA−IB, in terms of p, q, r.

Let A be an event, and
let IA be the associated indicator random variable: IA(ω)=1 if ω∈A,
and IA(ω)=0 if ω∉A. Similarly, let IB be the indicator of another
event, B. Suppose that, P(A)=p, P(B)=q, and P(A intersection
B)=r.
Find E[(IA−IB)2] in
terms of p,q,r?
2.Determine Var(IA−IB)
in terms of p,q,r?
The solution in Chegg
is for P(AUB)=r instead of P(A intersection B)=r. I need to know
how to find Var(IA−IB) in terms of p,q,r?

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.

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.

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.

Let △ ??? be a triangle. Prove that ? = 2? if and only if △ ???
is equilateral.

Let A=(0,0), B=(1,1), C=(-1,1), A'(2,0),B'(4,0), C'(2,-2).Show
that triangle ABC and triangle A'B'C' satisfy the hypothesis of
Proposition 2.3.4 in taxicab geometry but are not congruent in
it

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.

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