A Very Nice Geometry Problem | Math Olympiad | 2 Different Methods 

My method: Let H bet the height of 🔺️ABC H=3.sin44° Segment AB=H/sin22° AB=3.sin44°/sin22 Sin44°=2.sin22°.cos22° AB=6.sin22°.cos22°/sin22° AB=6.cos22° Since AB is 6.cos22°, 🔺️ABD must be rectangle at Â. So, angle BÂD=90° Angle BÂC = 114° Theta= 114°-90°= 24°

Yep, second method is much nicer than the first.

(theorem sines to the triangle ABC) AB : sin 44= 3: sin22 from which AB sin22 = 3*2 sin22 cos22 and AB= 6 cos22 i.e. BD cos22=AB therefore BAD=90° and BDA=68° since BDA= θ + 44 = 68 then θ=24°

Second metod - beautiful!!!

Thank you, very nice! I missed the similarity.

Let P, a point in AB so that angle BCP = 22° AC=3, BD=6, DC=x Obviously CP is bisector of < BCA. Triangles APC,ABC are similar => AP/AC=AC/AB => AP/3=3/AB⇒AB⋅AP=9 (1) Apply bisector theorem in triangle ABC : AP/AC=BP/BC=(AP+BP)/(AC+BC)=AB/(3+6+x) => AP/3=AB/(x+9) => AP=3AB/(x+9) (2) (1),(2) => AB⋅3AB/(x+9)=9 => AB²=3(x+9) => *AB²=3x+27* (3) Stewart’s theorem in triangle ABC : AB²⋅DC+AC²⋅BD=BC(AD²+BD⋅DC) 3(x+9)⋅x+9-6=(x+6)[AD²+6x] ………………………………………. -3(x²+3x-18)=(x+6)AD² -3(x+6)(x-3)=(x+6)AD² *AD²=-3x+9* (4) (3)+(4) => AB²+AD² =3x+27-3x+9 =36 AB²+AD² =36 However BD²=36 So *AB²+AD² = BD²* . *According to converse of the Pythagorean theorem* triangle ABD is orthogonal . In ∆ ABC : 22°+ 44° + 90° + θ = 180° => θ=24°

2nd method is a lot easier and smarter.Thank you professor.

Congrats! I always prefer the second method cause its a genuine geometrical aproach. Inthat exercise the first ethod also led us to use very usefull triigonometrics identies. Thanks.

By second method you can also say that angle BAD=90° since in triangle ABD E is mid point of BD and AE=BD/2

wrong image. Draw AE =3. Angle BAD is 90 degree, and DAC is 24 degree

@Math booster : 2cd solution is very Smart !!!!!

Just amazing!

The first method can be much simplified: (1) Law of sines in ABC: AB = 6 cos 22° (2) Law of cosines in ABD: AD² = 6² sin² 22° (3) Converse Pythagoras in ABD: ∠BAD = 90° (4) ∠BDA = 90° - 22° = 68° and θ = 68° - 44° = 24°

second method is the real method

Poorly formulated question. Which segment measures 6?

Manchmal habe ich das Gefühl, Sie halten uns für Idioten .... 2 mal 22 ist selbstverständlich 44 ... man braucht in dieser Stadien nicht 22 + 22 = 44 zu schreiben.