A Very Nice Geometry Challenge | 3 Different Methods to Solve

[@johnbrennan3372]

You can also use the sine rule to get a value for cos theta which is x+1/2x-2. Then using the cosine rule on the given angle theta you get cos theta= (x^2+4x)/ (2x^2 + 2x).Putting the two values for cos theta equal x=5.

[@holyshit922]

I did it the same way

[@marcgriselhubert3915]

H is the orthogonal projection of A on (BC), and we note simply theta = t In triangle ABH we have AH = AB.sin(t) = (X+1).sin(t), and in triangle ACH we have AH = AC.sin(2.t) = (X-1).2.sin(t).cos(t) sin(t)0, so we have X+1 = (X-1).2.cos(t), and we get cos(t) = (X+1)/(2.(X-1)) In triangle ABH we have BH = AB.cos(t) = (X+1).cos(t) = ((X+1)^2)/(2.(x-1)) In triangle ACH we have CH = AC.cos(2.t) = (X-1).(2.(cos(t)^2) -1) = (-X^2 +6.X-1)/(2.X-1)) after simplification.. Then as BH + CH = AB we have now: ((X+1)^2)/(2.(X-1)) + (-X^2 +6.X -1)/(2.(X-1)) = X, or 8.X/(2.(X-1) = X after simplification.. That gives 4.X = X^2 -X and so X = 5 (X0)

[@prime423]

Look at diagram. One angle is double the other. Bisecting the larger given angle is a start. Then use angle bisector information to solve. Analysis is the key to many solutions.!!

[@CanalMiTube]

Line from A to BC, it gets the point D. Angle BAD=θ. ABD y ADC are isosceles. So DC=1. Line from A perpendicular to DC, it gets the point E. By Pythagorean theorem: (AE)^2=(4x^2 -8x +3)/4. So, (AE)^2 + (BE)^2 = (AB)^2; (4x^2-8x+3)/4 +(x-1/2)^2=(x+1)^2. x=5. Greetings.

[@someone33342]

Amazing ,but how you calculated AE

[@prime423]

This is the famous 4-5-6 triangle most experienced Mathletes know!!There is also a very complicated trig solution.

[@arbenkellici3808]

I could find x in an easier way Using Sinus Theorem you can get cos theta Afterwards you use Cosinus Theorem and get another value for cos theta Putting the two values for cos theta you get x=5 You dont need two other methods They are exhausting Thank you professor!

[@ARN48411]

You always come up with great problems, puzzles. Thanks sir❤

[@misterenter-iz7rz]

Typical double angle puzzle, just draw a line to construct two isosceles triangles with angle t and 2t, all problems can be solved. 😊😊😊😊😊

[@prossvay8744]

X=5 units

[@yakupbuyankara5903]

X=5

[@JAMESYUN-e3t]

🎉

[@comdo777]

asnwer=1