In the previous section we saw the multiplication between a tangent and a chord. We also saw some solved examples. In this section we will see some details about incircles.
We have seen that, if we are given a circle, we can draw two tangents from any exterior point of the circle. See fig.32.38(b). It is shown again below:
• We now ask a reverse question:
Given two lines which meet at a point. Can we draw a circle such that, both the lines are tangents to the circle?
Let us analyse this situation:
1. Consider fig.32.53(a) shown below. The two red lines AB and AC meet at A
• We want a circle in such a way that AB and AC are it's tangents.
• The yellow circle with center at O1 is indeed such a circle
♦ This is because, the green radial lines from O1 are perpendicular to both AB and AC at the respective points of contact. So AB and AC are tangents
• Is any other circle possible in this way?
2. Consider the cyan circle (with center at O2) in fig.b.
• AB and AC are tangents to this circle also
♦ This is because, the green radial lines from O2 are perpendicular to both AB and AC at the respective points of contact. So AB and AC are tangents
• Is any other circle possible in this way?
3. Consider the magenta circle (with center at O3) in fig.c.
• AB and AC are tangents to this circle also
♦ This is because, the green radial lines from O3 are perpendicular to both AB and AC at the respective points of contact. So AB and AC are tangents
4. It is clear that, a large number of such circles are possible. So what is the criteria?
• To find the answer, we want to know the property which is common to all such circles.
• We can see that, the centres O1, O2, O3 etc., of all such circles lie on the 'bisector of the angle between AB and AC'.
• We can mark any point on the angle bisector.
♦ Draw a perpendicular to AB through that point.
♦ With the point as center, and the perpendicular distance as radius, draw a circle.
♦ Then AB and AC will be tangents to the circle.
• We can mark infinite number of points on the angle bisector. Each of those point will give a circle.
• So it is clear that infinite number of such circles are possible
• Now we ask another question:
Given three lines which meet at three different points to give a triangle. Can we draw a circle such that, all the three lines are tangents to the circle?
Let us analyse this situation:
1. Consider fig.32.54(a) shown below.
• The three red lines meet at A, B and C
• A blue circle is shown inside ΔABC.
♦ Radial lines from the center O are perpendicular to the red lines at the respective points of contact.
♦ So the blue circle is indeed our required circle. But how do we draw it? Let us try:
2. As before, we draw the 'bisector of the angle between AB and AC'.
• We try different points O1, O2, O3 etc., on that bisector.
• We can draw circles using all those points as center.
• All those circles will be tangential to AB and AC.
• But there is no guarantee that, the circle draw in this way will be tangential to BC also
3. Next we try another angle bisector. This time the one between AB and BC.
• The circles drawn are shown in fig.c.
• All those circles will be tangential to AB and BC.
• But there is no guarantee that, the circle draw in this way will be tangential to AC also
■ Thus we get an interesting result:
♦ The circles draw on the bisector between AB and AC are tangential to AB and AC
♦ The circles draw on the bisector between AB and BC are tangential to AB and BC
4. So consider the point of intersection of the two angle bisectors. It is marked as O in fig.32.55(a) below:
• From O, draw a perpendicular to AB.
• With O as center and the perpendicular distance as radius, draw a circle. This is shown in fig.32.55(b).
• All the lines AB, BC and AC will be tangential to the circle.
• This circle is called the incircle or inscribed circle of the triangle
5. Interestingly, the 'bisector of the angle between AC and BC' will also pass through the same point O. We can write this:
■ The bisectors of all the three angles of a triangle meet at a point
• Every triangle will have a circumcircle and incircle. See fig.32.56(a) below:
• But for quadrilaterals, it depends on the type:
♦ Some quadrilaterals have both circumcircle and incircle. Example: Square. Fig.32.56(b)
♦ Some quadrilaterals have only circumcircle but no incircle. Example: Rectangle. Fig.(c)
♦ Some quadrilaterals have only incircle but no circumcircle. Example: Rhombus. Fig.(d)
♦ Some quadrilaterals have neither circumcircle nor incircle. Example: Parallelogram. Fig.(e)
• While making engineering drawings, 'tangents of circles' have much significance.
• The fig.32.57 below shows a part of a screen shot of a cad program while drawing circles.
■ The red arrow shows the option 'Ttr'. That is., 'Tangent, tangent, radius'.
• If we give two tangents only, infinite number of circles are possible.
• This is clear from fig.32.53(c) that we saw above.
• So we must specify a radius also.
■ The green arrow shows the option 'TTT'. That is., 'Tangent, Tangent, Tangent'.
• If we give three tangents, then the unique circle can be clearly specified.
• This is clear from fig.32.55(b) above.
• Additional information about radius is not required.
In the next section, we will see the relation between the incircle and the sides of a triangle.
We have seen that, if we are given a circle, we can draw two tangents from any exterior point of the circle. See fig.32.38(b). It is shown again below:
 |
| Fig.32.38(b) |
Given two lines which meet at a point. Can we draw a circle such that, both the lines are tangents to the circle?
Let us analyse this situation:
1. Consider fig.32.53(a) shown below. The two red lines AB and AC meet at A
 |
| Fig.32.53 |
• The yellow circle with center at O1 is indeed such a circle
♦ This is because, the green radial lines from O1 are perpendicular to both AB and AC at the respective points of contact. So AB and AC are tangents
• Is any other circle possible in this way?
2. Consider the cyan circle (with center at O2) in fig.b.
• AB and AC are tangents to this circle also
♦ This is because, the green radial lines from O2 are perpendicular to both AB and AC at the respective points of contact. So AB and AC are tangents
• Is any other circle possible in this way?
3. Consider the magenta circle (with center at O3) in fig.c.
• AB and AC are tangents to this circle also
♦ This is because, the green radial lines from O3 are perpendicular to both AB and AC at the respective points of contact. So AB and AC are tangents
4. It is clear that, a large number of such circles are possible. So what is the criteria?
• To find the answer, we want to know the property which is common to all such circles.
• We can see that, the centres O1, O2, O3 etc., of all such circles lie on the 'bisector of the angle between AB and AC'.
• We can mark any point on the angle bisector.
♦ Draw a perpendicular to AB through that point.
♦ With the point as center, and the perpendicular distance as radius, draw a circle.
♦ Then AB and AC will be tangents to the circle.
• We can mark infinite number of points on the angle bisector. Each of those point will give a circle.
• So it is clear that infinite number of such circles are possible
• Now we ask another question:
Given three lines which meet at three different points to give a triangle. Can we draw a circle such that, all the three lines are tangents to the circle?
Let us analyse this situation:
1. Consider fig.32.54(a) shown below.
 |
| Fig.32.54 |
• A blue circle is shown inside ΔABC.
♦ Radial lines from the center O are perpendicular to the red lines at the respective points of contact.
♦ So the blue circle is indeed our required circle. But how do we draw it? Let us try:
2. As before, we draw the 'bisector of the angle between AB and AC'.
• We try different points O1, O2, O3 etc., on that bisector.
• We can draw circles using all those points as center.
• All those circles will be tangential to AB and AC.
• But there is no guarantee that, the circle draw in this way will be tangential to BC also
3. Next we try another angle bisector. This time the one between AB and BC.
• The circles drawn are shown in fig.c.
• All those circles will be tangential to AB and BC.
• But there is no guarantee that, the circle draw in this way will be tangential to AC also
■ Thus we get an interesting result:
♦ The circles draw on the bisector between AB and AC are tangential to AB and AC
♦ The circles draw on the bisector between AB and BC are tangential to AB and BC
4. So consider the point of intersection of the two angle bisectors. It is marked as O in fig.32.55(a) below:
 |
| Fig.32.55 |
• With O as center and the perpendicular distance as radius, draw a circle. This is shown in fig.32.55(b).
• All the lines AB, BC and AC will be tangential to the circle.
• This circle is called the incircle or inscribed circle of the triangle
5. Interestingly, the 'bisector of the angle between AC and BC' will also pass through the same point O. We can write this:
■ The bisectors of all the three angles of a triangle meet at a point
• Every triangle will have a circumcircle and incircle. See fig.32.56(a) below:
 |
| Fig.32.56 |
♦ Some quadrilaterals have both circumcircle and incircle. Example: Square. Fig.32.56(b)
♦ Some quadrilaterals have only circumcircle but no incircle. Example: Rectangle. Fig.(c)
♦ Some quadrilaterals have only incircle but no circumcircle. Example: Rhombus. Fig.(d)
♦ Some quadrilaterals have neither circumcircle nor incircle. Example: Parallelogram. Fig.(e)
• While making engineering drawings, 'tangents of circles' have much significance.
• The fig.32.57 below shows a part of a screen shot of a cad program while drawing circles.
 |
| Fig.32.57 |
• If we give two tangents only, infinite number of circles are possible.
• This is clear from fig.32.53(c) that we saw above.
• So we must specify a radius also.
■ The green arrow shows the option 'TTT'. That is., 'Tangent, Tangent, Tangent'.
• If we give three tangents, then the unique circle can be clearly specified.
• This is clear from fig.32.55(b) above.
• Additional information about radius is not required.
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