 Hello friends welcome again to yet another session on gems of geometry in the previous sessions We have discussed so many beautiful concepts and theorems and properties of circles pedal triangles Coaxial circles radical axis and many more so continuing with the trend. We are now going to discuss one one more very popular concept Related to triangles and circum circles and that's called Simpson's line So it's very famous now Simpson's lines is you know has been Very popular, but as a matter of fact it was first just particular line Which is now called a Simpson line and which will be discussed anyways was first discovered by Person called William Wallace in 1797 right so that's another fact that you know We have to just find out why it is called Simpsons line Okay, Simpsons line and as I told you it was discovered in 1797 Wow 1797 like almost 225 years now so Simpsons line 1797 But actually discovered by a person called William Wallace Actually, Simpson is this person who has done a lot of work in number theory mathematical geometry in other fields as well and one of his famous contribution is In Fibonacci series so he found out how Nth term in a Fibonacci sequence can be you know obtained using a Relationship yeah, okay, so now we will be discussing Simpsons line and what exactly Simpsons line We had we have just discussed in the previous session where we demonstrated using geojabra tool as to what exactly Simpsons line means so Let's discuss. Let's you know first state the you know the term and then we'll try to prove it So it says the feet of the perpendicular from a point to the sides of a triangle are co-linear If and only if the point lines on the circum circle now this co-linear points E F H Form a line and that line is called Simpsons line. Okay, so So what is the thing? So we have learned That in a given triangle if there is a point which is also called a pedal point and you drop three perpendiculars On to the three sides and you join the feet of the perpendicular So you will get something called a pedal triangle now This is a special case when the pedal point that is the point from which you drop the perpendicular lies on the circum circle of the Triangle right so you can see in this diagram D is the point and Of let's say ABC is the triangle and you can see there is a circum circle D is the point on the circum circle and from D D E D F and D G are the three perpendiculars dropped on to the three Sides, okay, and now we have to prove that E F G are Co-linear which is also called this line E F G is also called Simpsons line Okay, so let's try to prove it So here goes the proof Okay, and we will be using concepts already learned so far and not very difficult concepts So to say to prove this thing. Okay, so First let's consider trying There's a you know a B D C so a B D C is a cyclic quadrilateral Cyclic quadrilateral, isn't it cyclic quadrilateral? So So what does it mean it means? angle BDC BDC is equal to 180 degrees Minus angle a Okay, so angle B D C you can see angle B DC which is back angle B DC This is angle B DC so B DC is what 180 degree minus Angle a Okay. Now, if you consider quadrilateral in Quadrilateral quadrilateral which one E a G D E a G D E a G D, right? If you see angle E Is equal to 90 degrees in that quadrilateral isn't it and Angle D also is 90 degrees. I'm sorry not angle D angle G Angle G is 90 degrees. That means I can say therefore Angle a plus angle B. Sorry E D G is 180 degrees Clearly, right 180 degrees why because two opposite angles have a sum of 90 plus 90 that is 180 So the other pair of opposite angles should also be supplementary. I didn't mention the reason behind it You know this reason why what is this reason? This is nothing but in a cyclic quadrilateral opposite angles are Opposite angles are supplementary supplementary in a in a cyclic Quadrilateral, this is a reason, right? So now it's clear so angle a plus edg is also 180 degrees and hence angle edg Edg is also 180 degrees minus angle a Clear so from here These two things we can say angle B. DC is equal to angle edg Both are 180 degree minus a now B. DC if you look carefully is nothing but angle B D G plus Angle G DC angle G DC, right? And edg if you see is equal to angle edb Plus angle B D G Correct. Now if you see I can cancel this too. So what do I get? I get angle G DC is equal to angle edb Let me highlight these two angles in the diagram this angle. We are talking about this one This one is equal to This one and let's say both are equal to X Fair enough Right so far so good. So let's now proceed here and let me just take away these things. Okay now if you see B F D. So if you see angle B F D is equal to 90 degrees right and and Angle B ed is also 90 degrees Correct. Therefore we can say B Edf is a cyclic quadrilateral is a cyclic quadrilateral right the pair of opposite Sum of opposite angles is 180 degrees. So hence that particular quadrilateral will be a cyclic quadrilateral if that is so then angle E D B is equal to angle E F B Right and both are equal to X Once again edb, which is X is equal to E F B. Why this is nothing but angles in the same segment angles subtended subtended by angles up in by arc in the same segment in the same segment are equal Fair enough That's clear. So hence This angle is X Quick guys now if you see again I can say Which two one yeah, so I can again take G F D C yes, so if you if you see yeah, so Angle D F C Is equal to angle D G C is equal to 90 degrees both Are the art, you know perpendicular drop from the point D if you see so DFC is equal to DGC both are 90 degrees so what I'm talking about is this that this angle is 90 and This angle is also 90 right. So what does this mean this means again? D F G C and G and C are Cork are Either they form a what do you say the cyclic quadrature or we can say D F G and C are consicclic points Consicclic points Okay, why because you know consicclic points with let's say DC is the diameter DC is the diameter of that circle Right and you know that a diameter subtends 90 degrees at two points in any any point of the circle So DC is the diameter of that circle. Okay, therefore again if they are consicclic I can say angle G D C which is X is equal to G F C is equal to X both are X right what do I mean? If this angle is X so this angle has to be X So if you now look closely in the diagram, what do I see I see that angle E F B is equal to angle G F C and both are equal to X degrees correct and and BC B F C B F and C B F and C B F and C are collinear Right, so the so one side is collinear that that were automatically automatically if there are X both are equal to X therefore, what do we know and E F H or E F E F G must be collinear must be co-linear why simple logic BC is the straight line and F is a point on it and There are two angles, which are equal with F B and F C as one of the arms and The angle is such that the other arm are on the opposite sides of the two parts of BC So definitely E F N H or E F G sorry are collinear. What do I mean? I'll explain once again. So let's say there is a straight line Okay, this is a straight line and here is a point. So this point is my B This point is C and this point was F now. We have two points on the on either sides of BC such that This angle is X and this angle is also X That means this is acting as vertically opposite angles Right only vertically opposite angles could be equal like that that means when is vertically opposite angles possible That means the other let's say this was E and this was G. So E F G are also Co-linear or E G E F G fall on a Same straight line, right? So hence we could prove that E F G are Co-linear now this line E G is called Simpsons line This is what is the This is what is the Proof correct. So if you also look at it in this way that if D was anywhere else, but on the circum circle Then there the pedal triangle formed by the three feet that is triangle E F G Would have been non-zero, but the moment D falls on the circumcised circle the area of that triangle becomes Zero right so we say that in this case pedal triangle is degenerate D Generate Okay, so if you degenerate pedal triangle is there. Okay, that's what is this proof all about. I hope you understood the The definition or the description of the Simpsons line and it's proof