## Math 9 Chapter 2 Lesson 6: Properties of two intersecting tangents

## 1. Summary of theory

### 1.1. Theorem of two intersecting tangents

**Theorem**

If two tangents to a circle intersect at a point, then

- That point is equidistant from the two contacts
- The ray from that point passing through the center is the bisector of the angle formed by the two tangents
- The ray from the center passing through that point is the bisector of the angle formed by the two radii passing through the points of contact

– Angle formed by two tangents AB and AC is angle BAC

– Angle formed by two radii passing through the contacts is BOC

### 1.2. Circle inscribed in triangle

- The circle that touches the three sides of a triangle is called the incircle of the triangle, and the triangle is called the circumcircle.
- The center of the incircle of a triangle is the intersection of the interior bisectors of that triangle

### 1.3. The circle is tangent to the triangle

- The circle tangent to one side of a triangle and tangent to the extensions of the other two sides is called the circumcircle of the triangle. (The figure above is called the tangent circle in angle A of triangle ABC.)
- The center of the tangent in angle A is the intersection of the bisectors of the exterior angles B and C, or the intersection of the bisector of angle A with the bisector of angle B (or C).
- For a triangle with 3 circles tangent to the triangle

## 2. Illustrated exercise

### 2.1. Basic exercises

**Question 1: **Let (O) from M outside the circle draw two tangents MA and MB to (O). On ray OB take C such that OB=BC. CMR: \(\widehat{BMC}=\frac{1}{2}.\widehat{BMA}\)

**Solution guide**

We have: MO is the bisector of angle AMB, so \(\widehat{OMB}=\frac{1}{2}.\widehat{BMA}\)

Considering triangle OMC, whose OB is both the altitude and the median, the triangle MOC is isosceles at M, so MB is the bisector of angle OMC.

\(\Rightarrow \widehat{BMC}=\widehat{OMB}=\frac{1}{2}.\widehat{BMA}\)

**Verse 2: **Given a circle (O;R) and a point A outside the circle. Draw the tangent lines AB and AC. CMR: \(\widehat{BAC}=60^{\circ}\Leftrightarrow OA=2R\)

Solution guide

\(\widehat{BAC}=60^o\Leftrightarrow \widehat{OAB}=30^o\Leftrightarrow sin \widehat{OAB}=\frac{1}{2}=\frac{OB}{OA}=\ frac{R}{OA}\Leftrightarrow OA=2R\)

**Question 3:** Prove that the area of a triangle circumscribing a circle is calculated by the formula: S=pr, where p is the half-perimeter of the triangle, r is the radius of the inscribed circle.

**Solution guide**

\(S_{ABC}=S_{AOB}+S_{BOC}+S_{AOC}=\frac{1}{2}.(AB+BC+AC).r=pr\)

### 2.2. Advanced exercises

**Question 1:** Given a semicircle with center O and diameter AB. Draw the rays Ax perpendicular to AB, By perpendicular to AB on the same side as the semicircle. I is a point on the semicircle. The tangent at I intersects Ax, By at C, D respectively.

a) CMR: Triangle COD is a right triangle

b) Find the position of point I so that the perimeter of quadrilateral ACDB is the smallest. Calculate that circumference in terms of R

**Solution guide**

a) We have a right triangle IAB at I

Let E be the intersection of AI and CO, and F the intersection of IB and OD. Considering quadrilateral IEOF has 3 right angles, so IEOF is a rectangle deduced \(\widehat{EOF}=90^{\circ}\Rightarrow \Delta COD\) square at O

b) Since the tangent at A and I intersect at C, CA=CI, similarly DI=DB \(\Rightarrow AC+BD=CD\). We have \(CD\geq AB\) again because AB is the perpendicular segment of two parallel lines, AC and BD.

Then: \(2P_{ACDB}=AC+BD+CD+AB=2CD+AB\geq 3.AB=3R\)

**Verse 2: **Let ABC be an isosceles triangle at A inscribed in circle (O). The tangent lines of (O) drawn from A and C intersect at M. On the ray AM take D such that AD=BC. CMR: AC, BD, OM concurrent

**Solution guide**

We will first prove that ABCD is a parallelogram

We have AO perpendicular to BC, AO perpendicular to AD so \(AD\parallel BC\), and AD=BC so ABCD is a parallelogram

Let E be the intersection of AC and OM. According to the property of two intersecting tangents, E is the midpoint of AC (because triangle MAC is isosceles at M, with high ME).

Since ABCD is a parallelogram, the diagonal will pass through each line. So BD passes through E

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Given a circle (O), point A lies outside the circle. Draw the tangent lines AM, AN to the circle (M, N are the contacts)

a. Prove that OA MN

b. Draw the NOC diameter. Prove that MC // AO

c. Find the lengths of the sides of triangle AMN knowing OM = 3cm, OA = 5cm

**Verse 2: **Given a circle (O), the point M lies outside the circle. Draw tangent MD, ME to the circle (D, E are the contacts). Through I on the minor arc DE, draw a tangent to the circle, intersecting MD and ME at P and Q respectively. Knowing MD = 4cm, calculate the perimeter of triangle MPQ

**Question 3:** Given the angle xOy is different from the flat angle, the point A lies on the ray Ox. Construct a circle (I) passing through A and tangent to the two sides of angle xOy

**Question 4: **Given a semicircle with center O and diameter AB. Let Ax, By be the rays perpendicular to AB (Ax, By and the semicircle belong to the same half plane of edge AB). Let M be any point on the ray Ax. Through M draw a tangent to the semicircle, intersecting By at N.

a. Calculate the measure of angle MON

b. Prove that MN = AM + BN

c. Prove that AM.BN = R2 (R is the radius of the semicircle)

### 3.2. Multiple choice exercises

**Question 1: **Which of the following statements is correct?

A. There are 3 circles inscribed in a triangle

B. There is only one circle that is tangent to a triangle

C. The intersection of the interior bisectors is the center of the circle that is tangent to the triangle

D. the intersection of the bisector in angle A and the exterior bisector at B is the center of the circumcircle in angle A

**Verse 2:** Let triangle ABC be right-angled at A. O is the center of the circle inscribed in triangle ABC. D, E, F are the contacts on AB, AC, BC respectively. Which formula is correct?

A. AD=AC+AB-BC

B. 2AD=AB+AC-BC

C. 2EC=AB+AC-BC

D. 2BD=AC+BC-AB

**Question 3: **Given a circle (O). M is the point outside the circle, draw two tangents MA and MB of (O). Which of these following statements is wrong

A. \(MA=MB\)

B. \(OM\perp AB\)

C. \(\widehat{OMA}=\widehat{OMB}\)

D. \(OM=2.AB\)

**Question 4: **Let ABC be a right triangle at A with AB=3, AC=4. Circle (I;r) inscribed in triangle ABC. The value of r is:

A. 1

B. 2

C. 3

D. 4

**Question 5: **Let ABC be a triangle circumscribing the circle (O). Know \(\widehat{AOC}=130^o, \widehat{OCA}=30^o\). Compare OB and OC

A. OB

B. OB>OC

C. OB=OC

D. Not enough data for comparison

## 4. Conclusion

Through this lesson, you will understand some of the main topics as follows:

- Draw a circle inscribed in a given triangle.
- Apply properties of two intersecting tangents in calculation or proof exercises.
- Find the center of a circular object using the “bisector”.

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