Question

Дано:AB=BC; угол A =60°;CD-биссектриса угла BCE. Доказать:DC||AB.

Answer 1

А можно чертеж? я просто не понимаю, от куда E? это секущая?

Answer 2

Given Information:

We are given that AB = BC and angle A = 60°. CD is the bisector of angle BCE.

To Prove:

We need to prove that DC || AB.

Proof:

To prove that DC || AB, we can use the angle bisector theorem and the fact that AB = BC.

According to the angle bisector theorem, in triangle ABC, we have:

AC/BC = AD/BD

Since AB = BC, we can substitute BC with AB in the equation:

AC/AB = AD/BD

Since angle A = 60°, angle B = 180° - angle A - angle C = 180° - 60° - angle C = 120° - angle C.

Now, let's consider triangle ACD and triangle BCD. We have:

angle ACD = angle BCD (since CD is the bisector of angle BCE) angle ADC = angle BDC (vertical angles)

Using the angle-angle similarity criterion, we can conclude that triangle ACD and triangle BCD are similar.

Since they are similar triangles, the corresponding sides are proportional. Therefore, we have:

AC/AD = BC/BD

Substituting BC with AB, we get:

AC/AD = AB/BD

Since AB = BC, we can substitute BC with AB in the equation:

AC/AD = AB/BD

Since AB = BC, we can simplify the equation further:

AC/AD = AB/BD = 1

This implies that AC = AD and AB = BD.

Now, let's consider triangle ABD. We have:

AB = BD (proved above) angle ABD = angle ADB (since AD = BD)

Using the side-angle-side (SAS) congruence criterion, we can conclude that triangle ABD and triangle ADB are congruent.

Since they are congruent triangles, the corresponding angles are equal. Therefore, we have:

angle ABD = angle ADB

Since angle ABD = angle ADB, we can conclude that angle ABD is an isosceles angle.

In an isosceles triangle, the base angles are equal. Therefore, we have:

angle ABD = angle A

Since angle ABD = angle A, we can conclude that angle ABD is also 60°.

Now, let's consider triangle BCD. We have:

angle BCD = angle BDC (since CD is the bisector of angle BCE) angle BDC = angle ABD (proved above)

Using the angle-angle-angle (AAA) similarity criterion, we can conclude that triangle BCD and triangle ABD are similar.

Since they are similar triangles, the corresponding sides are proportional. Therefore, we have:

BC/BD = AB/AD

Since AB = BC, we can substitute BC with AB in the equation:

AB/BD = AB/AD

Since AB = BC, we can simplify the equation further:

AB/BD = AB/AD = 1

This implies that AB = BD and AB = AD.

Since AB = BD and AB = AD, we can conclude that triangle ABD is an isosceles triangle.

In an isosceles triangle, the base angles are equal. Therefore, we have:

angle ABD = angle ADB

Since angle ABD = angle ADB, we can conclude that angle ABD is also 60°.

Now, let's consider triangle ACD. We have:

angle ACD = angle BCD (proved above) angle ACD = angle ABD (proved above)

Using the angle-angle-angle (AAA) similarity criterion, we can conclude that triangle ACD and triangle ABD are similar.

Since they are similar triangles, the corresponding sides are proportional. Therefore, we have: