Diameter of Incircle Lemma and Dilation of Incircle
Suppose we have a triangle $ABC$. Let us extend the sides $BA$ and $BC$. We will draw the incircle of this triangle.
How to draw the incircle?
Here is the construction.
Draw any two angle bisectors, say of angle $A$ and angle $B$
Mark the intersection point $I$.
Drop a perpendicular line from I to one of the sides, say $AC$ in this picture.
Suppose the perpendicular intersects $AC$ at $E$.
The incircle is drawn centred at $I$ and with radius $IE$
Suppose EI intersects the incircle at F.
How to draw the excircle?
Now let us draw the excircle.
To do that we will need the angle bisector of external angle A and external angle C. Suppose they intersect at $I_A$. Drop a perpendicular from $I_A $ to extended $BA$ or extended $BC$ or $AC$. In this picture we drop it on extended $BA$ Suppose $J$ is the point of intersection of extended $BA$ and the perpendicular.
Draw a circle centred at $I_A$ and radius $I_A J$. This is the excircle.
Dilating the incircle to excircle
The incircle can be dilated or blown up with respect to point $B$ into the excircle. The center $I$ is sent to the center $I_A$ under dilation $FE$ which is perpendicular to $AC$ is sent to another segment perpendicular to $AC$ as angles are preserved under dilation
Questions:
What is the ratio of dilation?
How can you rigorously show that the center $I$ goes to the center $I_A$ under this dilation?
Where do the point $F$ go under this dilation?
Show that $AN = CE$
Geogebra workbook
Homothety 1
(This is a series of discussions on Homothety. It is largely derived from the Math Olympiad Classroom Discussion in Cheenta - cheenta.com)
Homothety is a geometric transformation. It has a couple of synonyms: dilation and central similarity.
A geometric transformation is a function. It can be thought of as a machine which takes in a point as input and gives out a point as output.
In algebra or calculus we encounter simple functions in school. One example is $ f(x) = x^2 $ . The rule of processing in this function is "square the input". That is if the input is 3, output will be 9. Geometric Transformations are similar things. It's set of input (often termed as "domain of the function") are however points instead of real numbers. Algebraically the points are doublets of numbers like (1,2) representing the coordinate of the point. Of course a point can also be represented by triplet of numbers (say (3, 5, -7) ) if we do geometry in three dimension (and similarly by a n-tuple $ (a_1 , a_2 , ... , a_n ) $ of numbers if we work in n-dimension) . Presently we shall confine our discussion in the plane or two dimension.
What kind of function (geometric transformation) is homothety?
To define homothety we need to specify two things: center of homothety and ratio of homothety. Center of homothety is a point on the plane. It can be any point. When we treat homothety algebraically, this point is regarded as the origin (0, 0) of the coordinate axis (assuming we are working in cartesian coordinate system). Ratio of homothety can be any real number positive, negative or zero.
Suppose O is the center of homothety and r is the ratio of homothety. If a point A on the plane is the input we can find the output point A' in the following manner: join OA, extend OA both sides and find a point A' on this line such that $\frac{OA'}{OA} $ = r .
In other words, the segment OA is stretched (or contracted) r times and the direction of this stretching or contraction is decided by the sign of r. Suppose r = 2, then OA will be stretched two times in the direction OA. However if r = -2, A' is on AO (on the other side of A about O) and length of OA' is twice of OA.
In the next installment of this series of articles we will continue with the applications of homothety.
(This is a series of discussions on Homothety. It is largely derived from the Math Olympiad Classroom Discussion in Cheenta - cheenta.com)