Question

К двум окружностям с центрами в точках о1 о2 касающимся внешним образом в точке А, проведена общая касательная В С (В и С-точки касания). Докажите, что угол ВАС-прямой.

Answer 1

использовано:сумма углов в треугольнике, перпендикулярность радиуса к касательной в точке касания, определение равнобедренного треугольника

Answer 2

The question you asked is about the geometry of circles and tangents. It is a common problem in mathematics and can be solved using some basic theorems and properties. Here is a detailed answer:

Given: Two circles with centers O1 and O2, touching externally at point A. A common tangent BC is drawn (B and C are the points of contact).

To prove: Angle BAC is a right angle.

Proof: Draw O1O2, the radii O1B and O2C to the points of contact B and C respectively, the common tangent AD perpendicular to O1O2, and the chords AB and AC forming angle BAC.

The tangents to a circle from a point are equal in length. Therefore, BD = AD = CD, and hence triangles BDA and CDA are isosceles and have equal base angles, i.e. angle ABD = angle DAB and angle DAC = angle CAD.

The sum of angles in a triangle is 180 degrees. From triangle BDA: angle BDA = 180 - 2 * angle DAB. From triangle CDA: angle CDA = 180 - 2 * angle DAC. Add these two equations. Angle BDA + angle CDA = 180 - 2 * angle DAB + 180 - 2 * angle DAC. Angle BDA and angle CDA are adjacent and their sum is 180 degrees. Simplify. 180 = 360 - 2 * (angle DAB + angle DAC); angle DAB + angle DAC = 90 degrees. Since angle BAC = angle DAB + angle DAC, then angle BAC = 90 degrees. This is what we wanted to prove.

You can also find more information and examples on this topic by searching the web. For instance, you can check out these links:

- [A solution to a similar problem](https://www.xn--80aaasqmjacq0cd6n.xn--p1ai/app/examples/view/Okruzhnost/Reshit-zadachu50/) - [An answer to another question](https://uchi.ru/otvety/questions/k-dvum-okruzhnostyam-s-tsentrami-v-tochkah-o1-o2-kasayuschimsya-vneshnim-obrazom-v-tochke) - [A presentation on circle geometry](https://infourok.ru/prezentaciya-reshenie-zadach-oge-po-matematike-po-teme-okruzhnost-i-ee-elementi-2630468.html)