At First Glance – 2400 Year Old Problem Solved: Philo's Line Construction With Unmarked Straightedge and a Compass

It looks like Philo’s Line can be drawn as below, using only a straightedge and a compass, by virtue of knowing NordaVinci’s Formula for the Area of a Right Triangle, which can be used to show that the two tangent points 1) the inscribed circle tangent point to the hypotenuse, and 2) the escribed circle tangent point to the hypotenuse, are equidistant from the endpoints of the hypotenuse on each end.

Dumbfounding! Amazing! and Flabbergasting!

“Look! Up in the sky! It’s a bird! No, it’s a plane! No, it’s another giant step forward in the history of trigonometry!

It comes right on the heels of NordaVinci’s Formula for the Area of a Right Triangle! We have just noticed, ladies and gentlemen, that not only the inscribed circle of a right triangle, but also the escribed circle of the same right triangle, are tangent to the hypotenuse at points that are equidistant from the center of the circumscribed circle of the same right triangle!

That might have been noticed before! But I didn’t notice it before! Now that I have, we also have to realize, that the two tangent points, one on the inscribed circle and one on the escribed circle are also the same distance from the two vertices of the right triangle that constitute the endpoints of the diameter of the circumscribed circle. (see picture)

Then there is the the further amazing fact of NordaVinci’s Formula for the Area of a Right Triangle being applied not only with respect to the tangent point of the inscribed circle, but also to the tangent point of the escribed circle!

Both tangent points serve the same function in that they divide the hypotenuse into two line segments, the product of whose lengths is the area of the right triangle!

Further interesting, and a previously recorded fact, is that the radius of the escribed circle opposite the right angle is equal to the inscribed circle radius plus the hypotenuse, which is none other than the famous semiperimeter of the right triangle, meaning obviously that the product of the inscribed radius and the escribed radius (the one just mentioned) is also equal to the area of the right triangle!

Dumbfounding! Amazing! and Flabbergasting!

Lost Theorem: Hint

I call it "NordaVinci's Formula." Here it is R(R + H) = X(H – X).

By the way, I discovered a new, previously completely unknown formula for the area of a right triangle. I call it “NordaVinci’s Formula.” Here it is R(R + H) = X(H – X). Here’s a link to my “NordaVinci’s Cat” poster with the Cat thinking about the derivation of NordaVinci’s Formula, that you can purchase for $20 – the poster, not the Cat. Be one of the first in the world to know something about the Right Triangle that is not in the Elements of Euclid Math Book or any other Math Book in Human History! Be one of the first to be able to brag to your trigonometry teacher that you know something that he or she doesn’t know about the area of a right triangle! Sell the poster to your teacher for only $30. Profits are for a non-profit group involving math, science and technology!

NordaVinci’s Formula is not simpler. That’s what kept it hidden for these many thousands of years – nobody cared, nobody needed to know it, and there was a simpler formula.

The interesting thing is that when I began to investigate, thinking that this must be common knowledge and I can’t imagine that I am the first person in recorded human history to notice this, I went to mathworld online reference and noticed in the footnotes the mention of Baker from 1884 who had collected a list of 130 different formulas for the area of a triangle, none of which is NordaVinci’s Formula.


Baker, M. “A Collection of Formulæ for the Area of a Plane Triangle.” Ann. Math. 1, 134-138, 1884.

I sent questions to some experts who didn’t bother to respond as of yet, and unless I pay somebody a few thousand dollars to do a literature search of extant scholarly journals, I won’t know if anyone in the world today ever wrote about this as something that was already discovered.

The brief point of NordaVinci’s Formula that was unknown is the right side of the equation. The left side was known already and easy to see from the diagram. The right side is simply X(H-X), which was unknown as the Area of a Right Triangle. That means that the tangent point of the inscribed circle to the hypotenuse, divides the hypotenuse into two line segments, of which the product of their lengths equals the area of the triangle.

That’s the actual contribution to the History of Trigonometry, that was lurking there just waiting to be discovered, waving its hand in the air, shouting, “Here I am!!!”

But nobody until me noticed it, apparently.

NordaVinci's Cat3

Spiritual Times of Glens Falls

[9/11 prophesied in 1985]

I have a spiritual son named Jon Hildreth in Glens Falls, NY, my hometown, where I now live. He was born in 1974, the year I joined the UC. He and his whole family are quite spiritual and attended the First Baptist Church of Glens Falls, the church that I also grew up in and taught Sunday School at when I was in the 12th grade.

Jon had a spiritual visitor who came into his room at night when he was growing up. His older sister, who was about 16 at the time that Jon was 10, claims that she could hear two voices through the wall separating her room from Jon’s. This visitor was not just a dream or a vision that only Jon could see and hear.

The visitor told Jon to wait until the two towers fell and that then he would meet a man named David, dressed in blue, who would give him a scroll.

Over the years when I was sometimes in my hometown, I would visit my First Baptist Church on Sunday. When I first joined, I prayed that Rev. A. King Wise who had pastored the church and had died in the 1960’s be able to study Divine Principle in the spirit world. In my dream at night I gave him a Divine Principle book. A couple years later, he returned in my dream and gave the book back to me, saying that he didn’t understand it.

Then in the late 1990’s I had a vision of the official big picture of Rev. Wise that remained in front of me for about 5 minutes. Taking this as a sign that he now understood Divine Principle, I attended more regularly the First Baptist Church.

Then after 9/11 while at a church meeting there, I met Jon Hildreth for the first time. He is the church historian. I had my blue suit on and a Divine Principle book in my hand and some of my comparative religion Tree of Life diagrams. He responded quite well and I gave him the Divine Principle book and told him that it might take him 40 years to accomplish his mission.

He is quite a spiritual brother as I said before, who also sees auras. So in conclusion, even though I am not named David, I was dressed in blue and gave him a scroll – the Divine Principle.

Also to point it out again, in case you missed it in the above narrative, Jon was 10 years old in 1985, when his spiritual visitor told him to “wait until the two towers fell, and then he would meet a man named David dressed in blue who would give him a scroll.” This was a prophecy given to him in 1985, while Rev. Moon Sung Myung was in Danbury Prison, that the World Trade Center would be destroyed.

Here’s a somewhat blurry photo of Jon and me at the First Baptist Church in Glens Falls NY around that time, with me wearing my blue suit.

Peter Nordquist with Jon Hildreth at the First Baptist Church of Glens Falls, NY

Rev. A.King Wise: Sermons

This book of Rev. Wise’s excellent sermons was compiled by Rev. Calvin C. Wilson after Rev. Wise’s passing on Sept. 18, 1967 (born 1920). Exactly 7 years later, Rev. Moon, Sung-Myung delivered the sermon “The New Future of Christianity” to over 20,000 people at Madison Square Garden.

As is sometimes the case with the premature departure to the eternal world of the spirit, Rev. Wise’s illness and death at 47, leaving his wife and young children behind, perhaps constituted some small indemnity condition for American Christianity to hear God’s message to revive America, delivered on that day by Rev. Moon. His message closed with an earnest request to pray and ask God directly to “know if Reverend Moon is telling the truth.”


Ladies and gentlemen, I have come here to Madison Square Garden tonight in obedience to God’s command. The Bible says: And in the last days it shall be, God declares, that I will pour out my spirit on all flesh, and your sons and daughters shall prophesy, and your young men shall see visions, and your old men shall dream dreams.’ (Acts 2:17)

We are living in such an extraordinary time, at the birth of a new age! Heaven is quite near. And if you earnestly call upon God, He will answer you.

You must urgently ask Him, “How can I know if Reverend Moon is telling the truth?” Do not let me or anyone else answer that question for you. Let God answer you directly.
So go in peace, and please ask God earnestly, sincerely. Confront God in prayer. God will reveal the answer to you.

The new hope for mankind is the Messiah. And that “great and terrible day of the Lord” is at hand. It is up to you whether that day will be great or terrible. If you meet the Messiah, for you that day will be great. But if you fail to meet him, then for you that day will indeed be terrible.

God bless you. Thank you for your attentive listening.

Why Would Conscience Be Higher Than Love?

I had another thought about the included diagram related to Rev. Moon having said that between “Love” and “Conscience”, it is Conscience that is a little bit higher than Love. Why would that be, exactly?

The answer might be related to the following observation from Rev. Moon’s words in the speech [9/23/1979]”Men of God:”

“For God, life comes first because first He existed and then He expressed love. But for man, love comes first. Love was the origin of man.”

[My Reflection]
Since God existed in the environment of the highest realm of heart which does not develop (identity maintaining 2 stage process centered on heart), before creating anything, and before having matured or grown up, then the highest realm of heart which never changes as the environment provided a standard that God’s Conscience used to compare to and to navigate along the path or train track of the Principle – thus Conscience is similar to if not identical to “the Autonomy of the Principle” that guides spiritual growth during the growth period, or the Indirect Dominion Through the Autonomy of the Principle.

Conscience thus allowed God to grow up to become the Creator God who creates through the two-stage process of the Developing 4-Position-Foundations [Inner and Outer] centering on the Purpose of Creation, which is . . . Love – such that there could even be such a thing as the 4th Stage of returning to one body with God, also known as The Direct Dominion of the Love of God.

Sakhalin, a Dream of Ancient Times, and Earthquakes

The first barrier to be broken in meditation is time, because mind transcends time and space. How does that work in practice? Here’s an example from my own experience.

One night I was wondering where some of my ancestors came from a long, long, long time ago in the far distant past. So I asked the question in prayer before going to sleep, and just before I woke up in the morning, I had an amazing dream that showed me the far distant past.

The dream began with me looking at a big atlas (book of maps of the world, for those of you who didn’t know that, or forgot). First I was looking at the Azores at a specific latitude and longitude – the remnants of the sunken continent of Atlantis.

I flipped the page and was looking at Sakhalin north of Korea. As I looked at Sakhalin, I saw a bow and arrow and a man who looked like an Aboriginee from Australia – one of my presumed ancestors from the far distant past. I could also see that at that time Sakhalin had a tropical climate – there were tropical plants and big flowers.

As the dream ended and I was about to open my eyes to wake up, I was literally, vocally saying the word “Sakhalin,” which I had never heard of before. I got up and looked it up and discovered that in fact the dream was correct in showing me the map of Sakhalin, which is really today named “Sakhalin.”

That very day Sakhalin was featured in the news, because there had been an earthquake in Sakhalin. “What a coincidence!” I thought. Was there a causal relationship between  my question, the dream in response, from which I had awoken with the word, “Sakhalin” on my lips – that I had never previously heard of or known about, and the earthquake!?

Do questions and prayers about the distant past  . . . cause earthquakes?!