Prove the given?

enter image source here

2 Answers
Jul 31, 2017

see explanation.

Explanation:

enter image source here
Given DeltaABC is an equilateral triangle,
=> angleABC=angleACB=angleBAC=60^@
Given PQ // AC
=> DeltaQBP and DeltaABC are similar
=> angleQBP=angleQPB=angleBQP=60^@
Let BP=x, => PQ=QB=x
Given CR=BP, => CR=x
angleQPM=180-60=120^@
angleRCM=180-60=120^@
=> angleQPM=angleRCM
Let angleQMP=y, => angleRMC=y

As angleQPM=angleRCM, angleQMP=angleRMC, and QP=RC,
=> DeltaQPM and DeltaRCM are congruent.
=> (QP)/(PM)=(RC)/(CM)

=> PM=CM

Hence, QR bisects PC at M. (proved)

Jul 31, 2017

See below.

Explanation:

Calling

A=(rho/2,rho sqrt3/2)
B=(0,0)
C=(rho,0)

we have

{(Q = B + lambda_1 (A - B)),( P = B + lambda_1 (C - B)),( R = A + (1 + lambda_2) (C - A)),( s_1 = P + mu_1 (C - P)),( s_2 = Q + mu_2 (R - Q)):}

Here 0 le lambda_i le 1 and 0 le mu_i le 1

To know if s_1 and s_2 intersect is necessary and sufficient that the equation

s_1 = s_2 or

P + mu_1 (C - P)=Q + mu_2 (R - Q)

have a solution with mu_1^@, mu_2^@ such that

0 le mu_i^@ le 1

but the system

P + mu_1 (C - P)=Q + mu_2 (R - Q)

after the pertinent substitutions reads

(((2 lambda_1-2) rho, (2 - lambda_1 + lambda_2) rho),(0, (lambda_1 + lambda_2) rho))((mu_1),(mu_2)) = ((lambda_1 rho),(lambda_1 rho))

and solving for mu_1,mu_2 we obtain

(mu_1^@,mu_2^@) = (lambda_1/(lambda_1 + lambda_2), lambda_1/(lambda_1 + lambda_2))
then as can we see

0 le mu_i^@ le 1 and then segments s_1 and s_2 intersect.