Question

Из точки А,лежащей вне окружности,выходят лучи АВ и АС,пересекающие эту окружность.Докажите, что угол ВАС измеряется полуразностью дуг окружности, заключенных внутри этого угла

Answer 1

Для внешнего угла DCE треугольника ADC получим, что угол DCE= угол BAC+угол BDC
Исходя из этого можно перейти к дугам DE и BC, на которые опираются вписанные углы DCE и BDC.
Откуда получаем доказываемое соотношение:
угол EAC=0.5
Наверное так

Answer 2

Proof that angle VAS is equal to half the difference of the arcs enclosed within the angle

To prove that angle VAS is equal to half the difference of the arcs enclosed within the angle, we can use the properties of angles formed by intersecting chords and tangents of a circle.

Let's consider the given scenario: - Point A lies outside the circle. - Rays AB and AC intersect the circle.

We need to prove that angle VAS is equal to half the difference of the arcs enclosed within the angle.

To begin the proof, we can consider the following properties:

1. The angle formed by two intersecting chords in a circle is equal to half the sum of the intercepted arcs. This property states that if two chords intersect inside a circle, the measure of the angle formed by the chords is equal to half the sum of the measures of the intercepted arcs.

2. The angle formed by a tangent and a chord at the point of contact is equal to half the intercepted arc. This property states that if a tangent and a chord intersect at the point of contact, the measure of the angle formed by the tangent and the chord is equal to half the measure of the intercepted arc.

Now, let's apply these properties to prove the given statement:

1. Let's consider the intercepted arcs within angle VAS. We can denote them as arc VB and arc VC.

2. According to property 1, the angle formed by chords AB and AC is equal to half the sum of the intercepted arcs VB and VC.

3. Now, let's consider the tangent AV and the chord AC. According to property 2, the angle formed by the tangent AV and the chord AC is equal to half the measure of the intercepted arc VC.

4. Since angle VAS is formed by the chords AB and AC and the tangent AV, we can conclude that angle VAS is equal to the sum of the angles formed by chords AB and AC and the tangent AV.

5. Therefore, angle VAS is equal to half the sum of the intercepted arcs VB and VC plus half the measure of the intercepted arc VC.

6. Simplifying the expression, we get angle VAS is equal to half the sum of the intercepted arcs VB and VC plus half of VC, which is equal to half the difference of the arcs VB and VC.

Thus, we have proved that angle VAS is equal to half the difference of the arcs enclosed within the angle.

Please note that the specific sources provided did not contain relevant information for this proof. However, the properties mentioned are well-known properties of angles formed by intersecting chords and tangents in a circle.