Circumcenter of a triangle.

[Pramod Kumar Tandon]

Equation of sides of a triangle are ------- 3x-y+8=0, 2x+y=3 and x+y=0. Without finding coordinates of the angular points, find coordinates of the circumcenter of the triangle. I have no clue how to go through.

 

[Harry_the_cat]

The circumcentre of a triangle is defined as the point where the perpendicular bisectors of the sides of that particular triangle intersect.
Draw a rough diagram. And roughly locate the circumcentre. Does that give you any hints??

 

[Pramod Kumar Tandon]

The circumcentre of a triangle is defined as the point where the perpendicular bisectors of the sides of that particular triangle intersect.
Draw a rough diagram. And roughly locate the circumcentre. Does that give you any hints??

Here is the exact diagram ? How should I proceed. The answer is (3,2)

 

[The Highlander]

Equation of sides of a triangle are ------- 3x-y+8=0, 2x+y=3 and x+y=0. Without finding coordinates of the angular points, find coordinates of the circumcenter of the triangle. I have no clue how to go through.

Hmmm. The "condition" specified (emboldened above) seems rather restrictive to me.
It is simple enough to find the gradients of the perpendicular bisectors (\(\displaystyle m_1 × m_2=-1\) which I assume you already know) but if you are not allowed to find the coordinates of the triangle's vertices then I am at a loss as to how you might find the equations of the perpendicular bisectors; maybe someone else can offer a solution as to how you find the intersection of three (non-parallel) lines when you only know their gradients?

Here is the exact diagram ? How should I proceed. The answer is (3,2)View attachment 33811

I presume that you constructed your diagram using the answer (3, 2) that was given for the circumcentre, ie: not an answer that you have arrived at yourself (hence the request: "How should I proceed")
The simplest way I can see to arrive at the answer is, having drawn (as previously advised) a diagram similar to the one below, it should be clear that, in this instance, finding the intersection of the red line (x+y=0) and the green line (3x-y+8=0) will provide the y-coordinate of the circumcentre whilst finding the intersection of the red line (x+y=0) and the blue line (2x+y=3) will provide the x-coordinate of the circumcentre but, since that entails finding the points A & B which are the "coordinates of the angular points" (or at least two of them), then apparently you are not allowed to do that!

​

A more "thorough" approach would be to find the midpoints of each side of the triangle and use them along with the gradients of the perpendicular bisectors to establish their equations; then finding the intersections of each pair would confirm the circumcentre's coordinates but, again, that (it seems to me) can only be done by finding the coordinates of the vertices!

Naturally, if you are already given the answer, then it is a simple matter to work 'backwards' from that extra piece of information to construct the complete diagram including all the salient points (as below) but it seems that you are still looking for some further advice on how to arrive at the circumcentre's coordinates without finding the coordinates of A, B or C!

​

I'm afraid I am not able to do that for you (maybe I'm missing something simple?) but perhaps someone else in the forum may well be able to make some suggestion(s) towards achieving that goal?

In the meantime, for the avoidance of any doubt (about exactly what the parameters of this problem are), please post the original question in its entirety including any diagrams that were supplied with it and the answer supplied if any (a picture would be ideal).

 

[Pramod Kumar Tandon]

I have now adopted an unusual method for solving this problem. I have tried to solve it at least for 3 times, but am getting wrong answer. My method is as follows -
If two lines with Equation E1 and E2 intersect, then equation of a third line passing through this point of intersection will be given by :
E1 + Φ E2 = 0. where Φ is a constant. Refer to the following figure -



Consider Joint A .
Any arbitrary line passing through A must have equation -
(3x-y+8) + Φ (2x+y-3) = 0
=> (3+2Φ) x + (Φ-1) y + (8-3Φ) = 0 ............... (1)
This line has slope = (3+2Φ)/(1-Φ)
This line will be perpendicular to AB, if it's slope = - 1/3
=> (3+2Φ)/(1-Φ) = - 1/3
=> Φ = -2
Hence Equation of this perpendicular line ( let's call it P ) will be : x(3-4) + y(-1-1) + (8+6) = 0
=> Equation of P : - x - 3y+14 = 0
=> x + 3 y = 14
Similarly Equation of Q will be obtained by keeping Φ=1/2
WE CAN, THUS FIND OUT EQUATION OF R,S,T and U
Now consider side AB. It has Two perpendiculars (P and R) at it's two ends. with known equations
These equations must differ by a constant. So we can find out equation of another perpendicular on the mid point of AB (shown by RED). It's equation must have constant = average of constant of P and R.
According to my calculations it comes out to be 9.
So equation of right bisector of AB will be : x+3y=9
Thus after knowing equation of any two right bisectors, Coordinates of circumcenter can be obtained.

 

[Pramod Kumar Tandon]

Now I have got the right Answer using this method after a lot of calculations.

 

[Deleted member 4993]

Equation of sides of a triangle are ------- 3x-y+8=0, 2x+y=3 and x+y=0. Without finding coordinates of the angular points, find coordinates of the circumcenter of the triangle. I have no clue how to go through.

Was this problem assigned to you as a homework in a class or you found it in a book as an excercize?

 

[Pramod Kumar Tandon]

Was this problem assigned to you as a homework in a class or you found it in a book as an excercize?

I composed it myself. Wanted a solution. Finally I succeeded. I usually love to play with mathematics.

 

[Pramod Kumar Tandon]

Hmmm. The "condition" specified (emboldened above) seems rather restrictive to me.
It is simple enough to find the gradients of the perpendicular bisectors (\(\displaystyle m_1 × m_2=-1\) which I assume you already know) but if you are not allowed to find the coordinates of the triangle's vertices then I am at a loss as to how you might find the equations of the perpendicular bisectors; maybe someone else can offer a solution as to how you find the intersection of three (non-parallel) lines when you only know their gradients?


I presume that you constructed your diagram using the answer (3, 2) that was given for the circumcentre, ie: not an answer that you have arrived at yourself (hence the request: "How should I proceed")
The simplest way I can see to arrive at the answer is, having drawn (as previously advised) a diagram similar to the one below, it should be clear that, in this instance, finding the intersection of the red line (x+y=0) and the green line (3x-y+8=0) will provide the y-coordinate of the circumcentre whilst finding the intersection of the red line (x+y=0) and the blue line (2x+y=3) will provide the x-coordinate of the circumcentre but, since that entails finding the points A & B which are the "coordinates of the angular points" (or at least two of them), then apparently you are not allowed to do that!

​

A more "thorough" approach would be to find the midpoints of each side of the triangle and use them along with the gradients of the perpendicular bisectors to establish their equations; then finding the intersections of each pair would confirm the circumcentre's coordinates but, again, that (it seems to me) can only be done by finding the coordinates of the vertices!

Naturally, if you are already given the answer, then it is a simple matter to work 'backwards' from that extra piece of information to construct the complete diagram including all the salient points (as below) but it seems that you are still looking for some further advice on how to arrive at the circumcentre's coordinates without finding the coordinates of A, B or C!

​

I'm afraid I am not able to do that for you (maybe I'm missing something simple?) but perhaps someone else in the forum may well be able to make some suggestion(s) towards achieving that goal?

In the meantime, for the avoidance of any doubt (about exactly what the parameters of this problem are), please post the original question in its entirety including any diagrams that were supplied with it and the answer supplied if any (a picture would be ideal).

I presume that you constructed your diagram using the answer (3, 2)
Sorry I got it constructed using Geo-gebra. I drew the circle after getting the three corner points using the software (circle through 3 given points)