*Vladimir Nikolin*

Let BH

_{b}and CH_{c}be two altitudes in ΔABC and O its circumcenter. As we know H_{b}H_{c}and AO are perpendicular. Attach:Geometry/Nigelstheorem.png Δ| **Figure 1**

### Another proof:

Let H' be the refection of H using line AB. We have, by the reflection property, AH=AH'. But, let H'' be the refection of H using line AC. We have, also by the reflection property, AH=AH'', so AH'=AH'', ΔH'AH'' is isosceles (Blue triangle - figure 1).

As we already know, points H' and H'' (and point A, of course) lies on the circumcircle of the triangle ABC. Its circumcenter is the point O as well and lies on the perpendicular bisector of the line segment H'H'' i.e. H'H'' is perpendicular to AO.

Finally, H

_{b}H_{c}is parallel to H'H'' (as midline of the triangle HH'H'') and we conclude that lines H_{b}H_{c}and AO are perpendicular.**Consequence 1:**

Let S be the area of the triangle ABC, R the circumradius and p

_{o}the semiperimeter of the orthic triangle H_{a}H_{b}H_{c}, p_{o}= (H_{a}H_{b}+ H_{b}H_{c}+ H_{c}H_{a})/2. Then**S = R·p**(only for acute triangles)

_{o} Attach:Geometry/Nigelstheorem2.png Δ| **Figure 2**

### Proof:

By Nagel's theorem, the sides of the orthic triangle are perpendicular to the radius-vectors from O to vertices of ΔABC. Which means that the quadrilaterals OH

_{a}CH_{b},OH_{b}AH_{c}and OH_{c}BH_{a}are orthodiagonal quadrilaterals, i.e. all have their diagonals perpendicular. The area of an orthodiagonal quadrilateral, convex or concave, with diagonals d_{1}and d_{2}is S = d_{1}·d_{2}/2. Now:- Area(ABC) = Area(OH
_{a}CH_{b}) + Area(OH_{b}AH_{c}) + Area(OH_{c}BH_{a})- = R·H
_{a}H_{b}/2 + R·H_{b}H_{c}/2 +R·H_{c}H_{a}/2

= R·(H_{a}H_{b}+ H_{b}H_{c}+ H_{c}H_{a})/2

= R·p_{o}

### Remark:

Since all antiparallels to a given side of a triangle are parallel to each other,

*Nagel's theorem*admits am immediate generalization: the line joining the circumcenter to a vertex is perpendicular to the antiparallels to the opposite side. There is a direct proof of that result.### References

- Johnson, R. A. Modern Geometry:An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929, p. 191