Olympiad question (prove)

[kerem2611]

Hi guys, I really need help with this question :/ (my sketch: https://www.geogebra.org/classic/abeyyk7p )

Let ABC be an acute, non-isosceles triangle with D is any point on segment BC. Take E on the side AB and take F on the side AC such that ∠DEB = ∠DFC. The lines DF, DE cut AB, AC at M, N, respectively. Denote (I1), (I2) as the circumcircle of DEM, DFN.

Let (J1) be the circle that internal tangent to (I1) at D and also tangent to AB at K, let (J2) be the circle that internal tangent to (I2) at D and also tangent to AC at H. Denote P as the intersection of (I1) and (I2) that differs from D and also denote Q as the intersection of (J1) and (J2) that differs from D.

(a) Prove that these points D, P, Q are collinear.

(b) The circumcircle of triangle AEF cuts the circumcircle of triangle AHK and cuts the line AQ at G and L (G, L differ from A).

(c) Prove that the tangent line at D of the circumcircle of triangle DQG cuts the line EF at some point that lies on the circumcircle of triangle DLG.

 

[Deleted member 4993]

Hi guys, I really need help with this question :/ (my sketch: https://www.geogebra.org/classic/abeyyk7p )

Let ABC be an acute, non-isosceles triangle with D is any point on segment BC. Take E on the side AB and take F on the side AC such that ∠DEB = ∠DFC. The lines DF, DE cut AB, AC at M, N, respectively. Denote (I1), (I2) as the circumcircle of DEM, DFN.

Let (J1) be the circle that internal tangent to (I1) at D and also tangent to AB at K, let (J2) be the circle that internal tangent to (I2) at D and also tangent to AC at H. Denote P as the intersection of (I1) and (I2) that differs from D and also denote Q as the intersection of (J1) and (J2) that differs from D.

(a) Prove that these points D, P, Q are collinear.

(b) The circumcircle of triangle AEF cuts the circumcircle of triangle AHK and cuts the line AQ at G and L (G, L differ from A).

(c) Prove that the tangent line at D of the circumcircle of triangle DQG cuts the line EF at some point that lies on the circumcircle of triangle DLG.

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem

 

[kerem2611]

I don't know how to prove a and c (b isn't a question idk why it says b)

 

[Steven G]

Your last post doesn't come close to answering Professor Khan's request which is Please show us what you have tried and exactly where you are stuck.

 

[kerem2611]

I know A B C D E F, How do I calculate to the formula of M1 and M2 algebraic? I need that first so I can get the coordinates of D and Q and then calculate the formula of DQ and DP so I can say the slopes are the same.

 

[Oliver-m]

Try sqrt(2)pi(x) = 2pi(x)/sqrt(2)
May help in geometry

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem

R

 

[kerem2611]

My bad, don't need to do it algebraic. Just geometry with coordinates. I want to prove that the angle of DQP is 180 degrees, because that would proof they're on a line. Someone that knows which steps to take to come to that?