Question

Prove the given?

Answer 1

see explanation.

Explanation:

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)

Answer 2

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.