- #1

Lemoine

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∠CED+∠ADE=∠CAD+∠ACE=∠A+∠C2=180°−∠B2

42°=180°−∠B2

∠B=96°

The other angles are difficult to calculate. I made this construction on Geogebra and got ∠A=12° and ∠C=72°(exactly), but I'm not being able to find them by geometric constructions. A teacher suggested me to drop perpendiculars from E to AD and from D to CE. Let E′ be the reflection of E over AD and D′ be the reflection of D over CE. Both D′ and E′ lie on AC, as AD bisects EE′ and CE bisects DD′.

What I found:

∠CED′=18°

∠ADE′=24°

∠DEE′=∠DE′E=90°−24°=66°

∠EDD′=∠ED′D=90°−18°=72°

∠E′ED′=66°−2⋅18°=30°

∠D′DE′=72°−2⋅24°=24°