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Given the base of a triangle and sum of its sides , is it possiible to tell the locus of circumcentre ,incentre , centroid ,orthocentre without using coordinate geometry ? My method was indeed using coordinate and setting up points coordinates as h,k to get the required locus like the incentre is going to be a ellipse . As i said i used coordinate so what i was expecting is a pure geometry solution , as such other solutions will most probably be using trigo, equations which i have already done . Updates :for incentre i found out that consider ABC triangle then observe that AB+ACis constant (assuming BC is the base), so A is moving in an ellipse now observe that AI/AD is a constant( D is the meeting point of AI line with BC) , hence incentre locus is an ellipse.

WizardMath
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  • Please include the results you obtained using coordinate geometry. This information could provide a helpful jumping-off for alternative approaches, and it will keep people from spending time duplicating your effort. – Blue Mar 01 '22 at 14:45
  • As i said i used coordinate so what i was expecting is a pure geometry solution , as such other solutions will be using trigo equations which i have already done – WizardMath Mar 01 '22 at 15:02
  • But yeah thanks for reminding i will add this to it @Blue – WizardMath Mar 01 '22 at 15:03
  • The easiest locus to determine is that of the circumcentre, which is the perpendicular bisector of the base of the triangles in question., – YNK Mar 01 '22 at 15:31
  • The locus of orthocentre doesn't look like a conic section. Is it possible to describe it without using coordinates? – Intelligenti pauca Mar 01 '22 at 18:46
  • Coordinate geometry came after classical Euclidean geometry. With invention of calculus, mathematicians further explored geometry to trigonometry (that can calculate for any angle) and differential geometry. Say, Pascal theorem (or so) seemed to be the limitation of classical Euclidean geometry. If Euclidean geometry was really self-contained to all purposes, mathematicians did not need to invent analytical geometry. In most cases of the special centres of a triangle, trilinear or barycentric coordinates might use and some might involve trigonometric functions. – Ng Chung Tak Mar 02 '22 at 03:42
  • For your further interest, see a dynamical (or instrumental) way to construct an ellipse from five points here which is already beyond the scope of compass-ruler construction. – Ng Chung Tak Mar 02 '22 at 03:43

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