Why was Jesus crucified on the cross?

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The essence of sin is the failure to love.  In their rebellion, Adam and Eve refused the “divine love made food” offered to them by God, preferring instead life on their own.  Cut off from loving communion with God, mankind has become enslaved to its own self-centeredness, the inevitable consequence of which is death.  God, however, was not willing that the creature which bore His image should simply return back to the nothingness from which he was created.  
Yet only one thing could accomplish this: a love stronger than man’s failure to love, ultimately, a love stronger than death.  
In our fallen existence, this profound self-emptying must inevitably take the form of sacrifice and suffering for such love is no longer natural to us.  This is why the Apostle Paul insists that the self-emptying of Christ was unto death, even the death of the Cross.  It was not enough that God simply appear on earth as man and teach us a better way to live.  For a man to be healed of his sin and achieve the purpose for which he was created, this “divine Love made flesh” had to enter into the lowest depths of human existence.  He had to partake, in His own person, of the ultimate consequences of man’s sinfulness.  
Man’s rebellion could be healed only by divine obedience.  Man’s self-centeredness could be destroyed only by the divine self-sacrifice on the Cross.  Man’s enslavement to death and corruption could be loosed only by the Resurrection of Him Who has life in Himself (John 5:26).  Finally, the limitations of man’s created nature could be transcended only by ascending to heaven with Him Who first descended to earth (cf. Ephesians 4:9-10).
In the Garden of Gethsemane, shortly before His arrest, Jesus prayed that the cup of suffering might pass from Him.  Nevertheless, he said, not as I will, but as you will. (Matthew 26:39).  
Thus, in the Garden of Gethsemane, Christ overcame the disobedience of Adam through His human obedience to the will of God the Father.  
After Jesus’ arrest and trial at the hands of the Jewish leaders and the Roman governor Pontius Pilate, He was beaten and mocked by the Roman soldiers.  They robed Him in purple, the color of royalty, and placed upon His most pure head a crown of thorns.  In this, Christ manifested the supreme humility of God and thereby conquered the pride of fallen man.
Upon the Cross, the love of God for mankind reached its apex as Christ descended to the lowest point of human existence.  
This self-emptying of God, for the sake of man, is the exact inverse of man’s attempt to assume for himself the glory of God.  By pouring out His most pure Blood upon the Cross, Christ not only blotted out the record of man’s sin, but overcame the power by which sin holds mankind captive.  Thus does the Cross of Christ, destroy the power of man’s rebellion.  
 
It is important to note that the suffering and death of Christ was effective for man’s salvation not merely because Christ was an “innocent” man unjustly slain, but because He was God.   Only God could take upon Himself the consequences of man’s sin and thereby destroy it.  Only God could enter the kingdom of death and fill it with His immortal life.  
The Resurrection of Christ frees all mankind from the bonds of corruption and death, because death had no power over Him Who is life and love Himself.
By assuming the entirety of our nature, Christ lived a perfect human life.  In becoming man, Christ recreated the original image of God in man, which had become distorted through sin.  
In rising from the dead, Christ brought with Himself the human nature that He had assumed and made it forever incorruptible.  Because of the consubstantiality of human nature, all men have a part in the resurrection of the dead.  
By ascending into heaven, Christ has taken our humanity and placed it forever at the right hand of God the Father.  Man was originally created to become like God and reign eternally with Him.     

"Go ye therefore, and make disciples of all the nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit"

Mathew 28:19