I can't see your complete topic, and I don't know. I found the textbook of the first volume of eighth grade mathematics in Baidu library, and I only answered the questions. Hehe, it's not easy. I hope it helps you.

1 1 problem proof

Because DE⊥AB of DF⊥AC

So ∠ AED =∠ AFD = 90.

And because AD shares ∠BAC.

So ∠EAD=∠FAD

And because AD is a common edge.

So Rt△AED≌Rt△AFD

So AE=AF

So △AEF is isosceles triangle.

So AD divides EF vertically in two.

12 problem proof

Because △ABC is an equilateral triangle and AD=BE=FC.

AF=AC-FC=BA-DA=BD

Because ∠A=∠B=60?

So △ ADF△ bed

So DF=ED

Similarly, it can be proved that FE=DF=ED.

So △DEF is an equilateral triangle

6-question proof

Because AD=BC, AC=BD, AB is the public party.

So △ ADB △ BCA

So ∠ABD=∠CAB

So △EAB is an isosceles triangle.

7-question proof

Because in Rt△ABC ∠ A = 30.

So BC= 1/2 AB.

Because ∠B is the angle, ∠ BDC = ∠ BCA = 90.

So Rt△BDC∽Rt△BCA

So ∠ BCD = ∠ A = 30.

So in Rt△BDC, BD= 1/2 BC.

So BD =1/2bc =1/2×1/2ab =1/4ab.