1. Dr. Mohamed BEN ALI

2.  By the end of this lecture, students will be able to: Understand the types of Tangents. Construct tangents. Construct incircle. Construct circumcircle Objectives

3.  Centre - the point within the circle where the distance to points on the circumference is the same.  radius - the distance from the centre to any point on the circle. The diameter is twice the radius.  circumference(perimeter) - the distance around a circle.  diameter - a chord(of max. length) passing through the centre Some definitions

4.  chord is a straight line joining two points on the circumference.  If line intersect the circle at two point that is called secant  tangent - a straight line making contact at one point on the circumference, such that the radius from the centre is at right angles to the line. Some definitions

5.  Given a circle, and any point on it, we can draw a straight line through the point that will be tangent to this circle

6.  Step 1: Draw the line through C and T C T

7.  Step 2: Draw a circle about T that passes through C, and let D denote the other end of that circle’s diameter C T D

8.  Step 3: Construct the straight line which is the perpendicular bisector of segment CD C T D tangent-line

9. Drawing a circle tangent to a line at a given point  At P, draw a perpendicular to the line AB  Set off the radius of the required circle on the perpendicular  Draw a circle with radius CP

10. Drawing tangents to two circles  Move the triangle and T-square as a unit until one side of the triangle is tangent, by inspection, to the two cirles,  Then slide the triangle until the other side passes through the centre of one circle, and lightly mark the point of tangency,  Then slide the triangle until the side passes through the centre of the other circle, and mark the point of tangency.  Finally, slide the triangle back to the tangent lines between the two points of tangency. Draw the second tangent line in a similar manner

11. Drawing two tangents circles Internally Tangent Externally Tangent

12.  Circle 1: centre C and radius R1  Circle 2: centre C’ and radius R2  Step 1: Draw the line through C and T  Step 2: Set your compass for radius R2  Step 3: Set your pointer on T.  Step 4: Make a mark on the right side of the line and label it C’  Step 5: Set your pointer on C’ and draw the circle 2. How do we do it? External tangent C T C’

13.  Circle 1: centre C and radius R1  Circle 2: centre C’ and radius R2  Step 1: Draw the line through C and T  Step 2: Set your compass for radius R2  Step 3: Set your pointer on T.  Step 4: Make a mark on the left side of the line and label it C’  Step 5: Set your pointer on C’ and draw the circle 2. How do we do it? Internal tangent C T C’

14. Drawing an arc tangent to a line or arc and through a point

17. Drawing an Arc Tangent to a Line and an Arc  Given line AB and arc CD AB C D  Strike arcs R 1 (given radius) R1R1 R1R1  Draw construction arc parallel to given arc, with center O O  Draw construction line parallel to given line AB  From intersection E, draw EO to get tangent point T 1, and drop perpendicular to given line to get point of tangency T 2 E T1T1 T2T2  Draw tangent arc R from T 1 to T 2 with center E

19. Drawing an arc tangent to two lines at Right Angles  Given two lines AB and BC with right angle ABC  With D and E as the points, strike arcs R equal to given radius A B C R R R  With B as the point, strike arc R equal to given radius O E D  With O as the point, strike arc R equal to given radius

21. Drawing an Arc Tangent to Two Lines at an Acute Angle A B C D  Given lines AB and CD  Draw parallel lines at distance R  Draw the perpendiculars to locate points of tangency  With O as the point, construct the tangent arc using distance R R R O

23. C D  Given lines AB and CD  Construct parallel lines at distance R  Construct the perpendiculars to locate points of tangency  With O as the point, construct the tangent arc using distance R R A B R O

25. Drawing an Arc Tangent to Two Arcs  Given arc AB with center O and arc CD with center S S D C O B A  Strike arcs R1 = radius R R1R1 R1R1  Draw construction arcs parallel to given arcs, using centers O and S  Join E to O and E to S to get tangent points T E T T  Draw tangent arc R from T to T, with center E R

29.  Given an ellipse, and any point on it, we can draw a straight line through the point that will be tangent to this ellipse F1F2

30.  Step 1: Draw a line through the point T and through one of the two foci, say F1 F1F2 T

31.  Step 2: Draw a circle about T that passes through F2, and let D denote the other end of that circle’s diameter F1F2 T D

32.  Step 3: Locate the midpoint M of the line- segment joining F2 and D F1F2 T D M

33.  Step 4: Construct the line through M and T (it will be the ellipse’s tangent-line at T) F1F2 T D M tangent-line

34.  Observe that line MT is the perpendicular bisector of segment DF2 (because ΔTDF2 will be an isosceles triangle) F1F2 T D M tangent-line

35. A B C o Construct a Δ ABC Bisect the side AB Bisect the side BC The two lines meet at O From O Join B Taking OB as radius draw a circumcircle.

36. A B C Construct a Δ ABC The two lines meet at O Taking OP as radius Draw a circumcircle Bisect the ABC Bisect the BAC Taking O draw OP AB O P