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). 

 

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.

 

[mathematic]

Draw a picture!

[kerem2611]

12 hours ago, mathematic said:

Draw a picture!

I already have, click on the link please

12 hours ago, mathematic said:

Draw a picture!

my sketch: https://www.geogebra.org/classic/hgyxdyxu

[mathematic]

The link appears to be a list of numbers - no picture?

[kerem2611]

number

[Country Boy]

On 4/24/2021 at 7:46 AM, kerem2611 said:

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. 

[/quote]

.

 

An acute scalene triangle.  Depending on exactly how "isosceles" is defined (it variies) saying a triangle is "non-isosceles" might include equilateral triangles

[A_curious_Homosapien]

On 6/1/2021 at 5:08 PM, Country Boy said:

An acute scalene triangle.  Depending on exactly how "isosceles" is defined (it variies) saying a triangle is "non-isosceles" might include equilateral triangles

It's clear that equilateral triangles are not included, because they are isosceles since the requirement of and isosceles triangle is that there should be atleast 2 equal sides and 2 equal corresponding angles which an equilateral triangles satisfies